Brago, King Eternal exiles everything simultaneously, and then returns everything to the battlefield simultaneously. With that out of the way:
1. The procedure for putting an Aura onto the battlefield without casting it is: First, choose an object to attach it to, then put it onto the battlefield attached to that object. Because the choice of object happens first and because everything is put onto the battlefield at the same time, you can't choose something that is also in exile and about to be put onto the battlefield. (Some cards like Warp World avoid this issue by explicitly putting enchantments onto the battlefield after everything else, but Brago doesn't do this.)
2. Sure. You don't put triggered abilities onto the stack until the current resolving ability (Brago's) is finished resolving, so by the time you can choose targets for Felidar Guardian's ability, everything is back on the field.
3. Anything that is a permanent card can be put back onto the battlefield (face up of course.) If you blink a manifested instant or sorcery card, it will remain in exile. This is because there is a rule that explicitly prohibits instant or sorcery cards from ever being put onto the battlefield; you simply ignore any instruction that tells you to and leave the card in the zone it was in.
Stack: Anything that triggers as a result of something that happens during the resolution of Brago's ability (generally, enters-the-battlefield and leaves-the-battlefield abilities) will wait to be put onto the stack until Brago's ability is done resolving. Then, players put triggers they control onto the stack in APNAP order, starting with you since you're the active player. If an opponent has abilities that trigger, theirs will end up on the stack above yours and will resolve first. If a single player controls multiple abilities that need to be put onto the stack at the same time, they put them there in whatever order they choose.
No. Both of those cards give you an alternative to paying the card's mana cost, but the "commander tax" is not part of the mana cost - it is an additional cost. You factor in additional costs after applying alternate costs.
It's worth noting that attaching an Aura to something this way doesn't target anything. (Auras only have targets while they are spells on the stack.) If an effect causes you to put an Aura directly onto the battlefield, you just choose something for it to enter attached to. So you could even attach the Aura to a creature with shroud, or an opponent's creature with hexproof, if you want.
Dusk Urchins counts the number of -1/-1 counters it had on it when it died. Not the number that was necessary to reduce its toughness to exactly zero. Also, you're definitely getting all of those -1/-1 counters because (a) Skinrender's ability puts them all on at once, and (b) even if it didn't, as you note, Urchins doesn't die until SBAs are checked after the ability has finished resolving.
Rules definition of this part of how protection works:
702.16c A permanent or player with protection can’t be enchanted by Auras that have the stated quality. Such Auras attached to the permanent or player with protection will be put into their owners’ graveyards as a state-based action. (See rule 704, “State-Based Actions.”)
702.16d A permanent with protection can’t be equipped by Equipment that have the stated quality or fortified by Fortifications that have the stated quality. Such Equipment or Fortifications become unattached from that permanent as a state-based action, but remain on the battlefield. (See rule 704, “State-Based Actions.”)
Auras, Equipment, and Fortifications as described here are all types of permanents. An object can never be a spell and a permanent at the same time, so spells can't ever invoke these two rules.
A spell is one of the following things:
1. A card that is on the stack, or an object on the stack that was originally a copy of a card in another zone and which was then cast (e.g. via Isochron Scepter's ability), or
2. An object on the stack that is a copy of a spell.
Refuse ceases to be a spell after it finishes resolving and goes to your graveyard. By the time you have the opportunity to cast Cooperate, there is no Refuse spell that can be targeted anymore.
Yes. An object with no mana cost has a converted mana cost of 0.
(There are entire decks built around the idea of cascading into cards like Living End or Restore Balance.)
To summarize the rules quote: If two people want to keep one-upping each other by taking actions they could repeat any number of times, the active player is the one who eventually has to do something different (generally, by not continuing to do that thing.) So if it's your turn and your opponent wants to keep skipping the same number of turns you're trying to skip, you're the one who has to stop trying. If it's your opponent's turn, they have to stop counter-activating the Vapors.
So tactically, your best option is to wait until your opponent's turn. Of course, if you get Vapors onto the field the usual way (i.e. during your main phase) then this creates a window where your opponent can destroy it before it's their turn, if they anticipate you wanting to use it to skip an arbitrary number of turns.
Do equipments and auras attached to permanents i control phase out directly or inderectly, since teferi's protection say all your permanents phase out? Does this change anything about the equipments and auras phasing in attached to those permanents? Just want to make sure i can play this in my equip deck...
Indirectly. There's actually an explicit rule for this:
702.25g If an object would simultaneously phase out directly and indirectly, it just phases out indirectly.
Note, the only time where this really makes a difference is if you have an Aura or Equipment attached to a token. Phased-out tokens cease to exist as a SBA, thus can't phase in, and thus anything that phased out indirectly because it was attached to it can't phase in either.
Nope, you're still good. Again, phasing doesn't cause anything to leave or reenter the battlefield; it's still the same object, so continuous effects that applied to it before will still apply to it. (In the case of Control Magic, this is also covered by the indirect phasing mechanism that Rezzahan described.)
Think of a permanent phasing out as though it's holding up a sign that says "I'm not here. Pay no attention to me." The game will oblige until it puts the sign down (phases in), then it will act like nothing happened.
One corner case/exception: If a control-changing effect (or any continuous effect, really) has a stated duration of "for as long as you control this permanent" (e.g. Willbreaker) and that permanent phases out, the effect will end because the game can't find that permanent.
You are correct for the reason you expected. If Walking Ballista has no counters on it, it isn't possible to remove one. And if you're unable to pay the cost to activate an ability, you can't activate it.
You get an extra mana for each Regal Behemoth. The "while you're the monarch" clause means it only triggers if you're the monarch, but otherwise it works the same as any other card with a triggered mana ability.
In order to deal combat damage, a creature must be on the battlefield and have the status of attacking or blocking when the Combat Damage step begins. If the creature is removed from the battlefield (or, less commonly, removed from combat*) before then, it won't deal any damage.
There are a few situations where something like Celestial Flare could be useful to cast after combat damage has been dealt (perhaps because blocking creatures have destroyed all but one attacker, and the defending player wants to make sure that that particular attacking creature is the one that gets sacrificed) and it is possible to cast Flare at this point because there are a couple of steps when creatures continue to count as attacking/blocking after damage has been dealt. This might be why you got varying answers.
*Most typically, creatures are removed from combat as a result of regenerating; if an attacking Cudgel Troll is targeted by Doom Blade and its regeneration ability is activated in response, it will be removed from combat as a result of regenerating. Some one-shot effects like Hollowhenge Spirit can explicitly remove a creature from combat as well.
This means that whoever designed those cards really meant for cards with Whispersilk Cloak to get only the quality of “can’t be blocked”, and really meant for Ogre Marauder to gain the ability, and not only the quality.
I can only speculate as to the intentions of the rules and development teams, but I suspect that most of the time, the interaction with ability-removing effects isn't the primary motivation for how a card is worded. I find it interesting that Ogre Marauder's printed text would have caused it to get the can't-be-blocked property without adding an ability just to get errata where it does add an ability, but likely someone decided that that was a cleaner way to do it within the rules. (My best guess: Making it add an ability is cleaner because otherwise, "can't be blocked unless defending player sacrifices a creature" could be interpreted as a cost imposed on blocking rather than a thing that can be done to prevent the triggered ability from having its primary effect.)
but if anyone show me a card giving only the quality of a keyword ability, then I’ll happily agree with the explanations already given.
This will be hard to find, because from a practical perspective, if something added a characteristic that did exactly the same thing as a keyword ability, it would be simpler to just add the keyword ability. I can think of several cases where a card does something that is very close to the same but not quite, such as: Vines of Vastwood being similar to hexproof (but not quite the same since it works differently if targeting an opponent's creature), or Spirit Link being similar to lifelink (but not the same because it's a triggered ability).
The main point here is that the property of "can't be blocked" isn't inherently an ability; it's just something that's true about the creature, and the important thing is to note what is causing that to be the case. If the thing granting "can't be blocked" status is an ability on the creature (either printed on it or explicitly the effect of an ability added to it, as with Ogre Marauder) then that ability can be removed. If it's a static ability of another permanent (Whispersilk Cloak) or the result of a one-shot effect (e.g. Distortion Strike) then ability-removing effects that affect the creature won't do anything. This is true for the other cases earlier; Sudden Spoiling won't do anything to the effect of Vines of Vastwood or Spirit Link either.
It's worth noting that Indestructible used to work the same way, before it was a keyword ability. Sudden Spoiling would make Darksteel Colossus stop being indestructible because it removed the ability "Darksteel Colossus is indestructible" but it wouldn't remove the status if it were granted by Shield of Kaldra. Now that Indestructible is a keyword ability, its interaction with ability-removing effects is more intuitive. I can't speak for the dev team on why they didn't make "unblockable" a keyword ability as well (though my guess is, there's design space around "can't be blocked except" and "can't be blocked unless" that they didn't want to close off.)
1. The procedure for putting an Aura onto the battlefield without casting it is: First, choose an object to attach it to, then put it onto the battlefield attached to that object. Because the choice of object happens first and because everything is put onto the battlefield at the same time, you can't choose something that is also in exile and about to be put onto the battlefield. (Some cards like Warp World avoid this issue by explicitly putting enchantments onto the battlefield after everything else, but Brago doesn't do this.)
2. Sure. You don't put triggered abilities onto the stack until the current resolving ability (Brago's) is finished resolving, so by the time you can choose targets for Felidar Guardian's ability, everything is back on the field.
3. Anything that is a permanent card can be put back onto the battlefield (face up of course.) If you blink a manifested instant or sorcery card, it will remain in exile. This is because there is a rule that explicitly prohibits instant or sorcery cards from ever being put onto the battlefield; you simply ignore any instruction that tells you to and leave the card in the zone it was in.
Stack: Anything that triggers as a result of something that happens during the resolution of Brago's ability (generally, enters-the-battlefield and leaves-the-battlefield abilities) will wait to be put onto the stack until Brago's ability is done resolving. Then, players put triggers they control onto the stack in APNAP order, starting with you since you're the active player. If an opponent has abilities that trigger, theirs will end up on the stack above yours and will resolve first. If a single player controls multiple abilities that need to be put onto the stack at the same time, they put them there in whatever order they choose.
Dusk Urchins counts the number of -1/-1 counters it had on it when it died. Not the number that was necessary to reduce its toughness to exactly zero. Also, you're definitely getting all of those -1/-1 counters because (a) Skinrender's ability puts them all on at once, and (b) even if it didn't, as you note, Urchins doesn't die until SBAs are checked after the ability has finished resolving.
Rules definition of this part of how protection works:
Auras, Equipment, and Fortifications as described here are all types of permanents. An object can never be a spell and a permanent at the same time, so spells can't ever invoke these two rules.
1. A card that is on the stack, or an object on the stack that was originally a copy of a card in another zone and which was then cast (e.g. via Isochron Scepter's ability), or
2. An object on the stack that is a copy of a spell.
Refuse ceases to be a spell after it finishes resolving and goes to your graveyard. By the time you have the opportunity to cast Cooperate, there is no Refuse spell that can be targeted anymore.
(There are entire decks built around the idea of cascading into cards like Living End or Restore Balance.)
So tactically, your best option is to wait until your opponent's turn. Of course, if you get Vapors onto the field the usual way (i.e. during your main phase) then this creates a window where your opponent can destroy it before it's their turn, if they anticipate you wanting to use it to skip an arbitrary number of turns.
Note, the only time where this really makes a difference is if you have an Aura or Equipment attached to a token. Phased-out tokens cease to exist as a SBA, thus can't phase in, and thus anything that phased out indirectly because it was attached to it can't phase in either.
Think of a permanent phasing out as though it's holding up a sign that says "I'm not here. Pay no attention to me." The game will oblige until it puts the sign down (phases in), then it will act like nothing happened.
One corner case/exception: If a control-changing effect (or any continuous effect, really) has a stated duration of "for as long as you control this permanent" (e.g. Willbreaker) and that permanent phases out, the effect will end because the game can't find that permanent.
You are correct for the reason you expected. If Walking Ballista has no counters on it, it isn't possible to remove one. And if you're unable to pay the cost to activate an ability, you can't activate it.
There are a few situations where something like Celestial Flare could be useful to cast after combat damage has been dealt (perhaps because blocking creatures have destroyed all but one attacker, and the defending player wants to make sure that that particular attacking creature is the one that gets sacrificed) and it is possible to cast Flare at this point because there are a couple of steps when creatures continue to count as attacking/blocking after damage has been dealt. This might be why you got varying answers.
*Most typically, creatures are removed from combat as a result of regenerating; if an attacking Cudgel Troll is targeted by Doom Blade and its regeneration ability is activated in response, it will be removed from combat as a result of regenerating. Some one-shot effects like Hollowhenge Spirit can explicitly remove a creature from combat as well.
This will be hard to find, because from a practical perspective, if something added a characteristic that did exactly the same thing as a keyword ability, it would be simpler to just add the keyword ability. I can think of several cases where a card does something that is very close to the same but not quite, such as: Vines of Vastwood being similar to hexproof (but not quite the same since it works differently if targeting an opponent's creature), or Spirit Link being similar to lifelink (but not the same because it's a triggered ability).
The main point here is that the property of "can't be blocked" isn't inherently an ability; it's just something that's true about the creature, and the important thing is to note what is causing that to be the case. If the thing granting "can't be blocked" status is an ability on the creature (either printed on it or explicitly the effect of an ability added to it, as with Ogre Marauder) then that ability can be removed. If it's a static ability of another permanent (Whispersilk Cloak) or the result of a one-shot effect (e.g. Distortion Strike) then ability-removing effects that affect the creature won't do anything. This is true for the other cases earlier; Sudden Spoiling won't do anything to the effect of Vines of Vastwood or Spirit Link either.
It's worth noting that Indestructible used to work the same way, before it was a keyword ability. Sudden Spoiling would make Darksteel Colossus stop being indestructible because it removed the ability "Darksteel Colossus is indestructible" but it wouldn't remove the status if it were granted by Shield of Kaldra. Now that Indestructible is a keyword ability, its interaction with ability-removing effects is more intuitive. I can't speak for the dev team on why they didn't make "unblockable" a keyword ability as well (though my guess is, there's design space around "can't be blocked except" and "can't be blocked unless" that they didn't want to close off.)