Yes they can. First off, Sigarda only affects humans on the battlefield, not human spells on the stack. Second, hexproof only works if the card is on the battlefield, not the stack or any other zone, so a spell with hexproof can still be targeted by a counter spell and can be countered like normal.
Assuming it's your opponent's turn, after their planeswalker resolves, they get priority first, not you. They will be able to activate one of the loyalty abilities before you get priority to activate Chainer's ability.
It's not an "infinite" loop, because it has an end: either their opponents will be dead due to the Tremors reducing their life total to 0, or they deck themselves.
After they've drawn a card, the God's ability triggers, and when that trigger resolves, they create a token. The token enters, and both the Discovery and Tremors trigger. Since they control both triggers, they get to choose the order they're put on the stack. More than likely, they're going to want to put the Tremors trigger on the stack last (so it resolve first), and the opponents are dealt damage before they draw their next card (and the loop starts again). State-based actions (which is what kill players) are checked after each trigger resolves. Assuming their opponents have a lower life total then the number of cards in their library, they will kill their opponents (if one or more have a higher life total, or they stack their triggers wrong so they draw before the Tremors trigger deals damage, then they'll lose due to drawing from an empty library (since the Discovery's trigger is not optional) before they kill their opponent).
For example, if I cast Casualties of War targeting an Eldrazi Monument and a Runeclaw Bear, would the Bear end up getting destroyed because the ruling for Casualties of War states that they are destroyed in the order specified? Thus (since artifacts are listed first) Eldrazi Monument will get destroyed first and stop granting indestructibility to Runeclaw Bear in time for Casualties to destroy it?
You follow the instructions in the order they're printed on the card. The artifact is destroyed first, then the creature, so the Monument is destroyed, then the Bear (which is no long indestructible).
Does the same hold true for Decimate?
No, Decimate is not the same. Everything is destroyed at the same time with Decimate (because it only uses the word "destroy" once), so targeting the Monument and the Bear with Decimate would mean that the Bear survives. Casualties of War uses "destroy" multiple times, so they're destroyed one at a time, not all at once.
Yes. It's still your commander, even if it's face down, so its combat damage counts towards the 21 or more points of combat damage that causes a player to lose the game.
What colors of mana you use to activate the ability does not matter. What matters is what colors the source of the ability is. All of the guildmages are multicolored, so they can target the Sphinx with their abilities.
No. You don't get to take the Spawn's triggered ability into account when determining if a creature has been assigned what would be lethal damage or not. You have to assign damage like normal, you can't assign one damage to each blocker (unless they're X/1s).
Snow-Covered Lands are not legal in standard. They are not printed in a standard legal set. So no, they could not play them in a standard tournament and would be forced to exchange them for non-snow variants. And yes, there are cards in standard that would care (Unmoored Ego).
And in limited, you can only add the main basic lands to your deck - you cannot add their snow-covered equivalents to your deck (if you want to play snow-covered lands in your draft deck, then you need to draft them). From the Magic Tournament Rules, Section 7.2 on Card Use in Limited Tournaments:
Players may add an unlimited number of cards named Plains, Island, Swamp, Mountain, or Forest to their deck and sideboard. They may not add additional snow basic land cards (e.g. Snow-Covered Forest, etc) or Wastes basic land cards, even in formats in which they are legal.
There's only one combat phase during the turn. A creature with double strike doesn't attack twice and it doesn't give you two combat phases - it can just deal combat damage twice during combat. If they want to attack with the vehicle, then it has to be declared as an attacker at the same time as other creatures are declared as attackers. That means if they're relying on the double strike creature to crew the vehicle, they won't be able to both crew the vehicle and attack with the double strike creature (since they'd have to crew the vehicle before declaring attackers, tapping the creature with double strike and preventing it from attacking). And vigilance just prevents it from becoming tapped as part of declaring attackers. If it's tapped, then it can't be declared as an attacker.
No. Gaining a basic land type doesn't also give it the Basic supertype. Mystic Compass would also not let your land dodge Dust Bowl's ability. The two main differences between the cards are:
A. Mystic Compass can target any player's lands; Navigator's Compass can only target your own lands.
B. Mystic Compass doesn't let the targeted land keep its other types, so the land would lose all other abilities and just have the ability to tap for mana of the chosen color. Navigator's Compass has it keep its other types, so it would just be gaining an ability instead of only having that ability.
The choice of whether or not to pay (and generating the treasure if they don't) happens as part of the triggered ability resolves. You do not get priority between them choosing not to pay and the treasure being created. If you cast it before the trigger resolves, then they can float mana to pay for the trigger when it resolves. If you cast it after the trigger resolves, then the treasure you just created will be exiled.
The 6/6 does not have trample (creatures in a band do not share abilities), so it has to assign all of its damage to the defending 5/5. The 4/4 with trample can assign all 4 of its damage to the defending player.
When assigning damage, you can take into account damage being assigned by other creatures at the same time into account to determine if it has lethal damage assigned to it (this is true in general, not just with banding). So the 4/4 trampler can see the 6/6 has assigned what would be lethal damage to the 5/5, and can assign all of its damage to the defending player.
510.1c A blocked creature assigns its combat damage to the creatures blocking it. If no creatures are currently blocking it (if, for example, they were destroyed or removed from combat), it assigns no combat damage. If exactly one creature is blocking it, it assigns all its combat damage to that creature. If two or more creatures are blocking it, it assigns its combat damage to those creatures according to the damage assignment order announced for it. This may allow the blocked creature to divide its combat damage. However, it can’t assign combat damage to a creature that’s blocking it unless, when combat damage assignments are complete, each creature that precedes that blocking creature in its order is assigned lethal damage. When checking for assigned lethal damage, take into account damage already marked on the creature and damage from other creatures that’s being assigned during the same combat damage step, but not any abilities or effects that might change the amount of damage that’s actually dealt. An amount of damage that’s greater than a creature’s lethal damage may be assigned to it.
The stack resolves one spell or ability at a time, not all at once. After each spell or ability resolves, everyone gets priority again before the next spell or ability resolves. Each spell will resolve one at a time, not all at the same time.
After they've drawn a card, the God's ability triggers, and when that trigger resolves, they create a token. The token enters, and both the Discovery and Tremors trigger. Since they control both triggers, they get to choose the order they're put on the stack. More than likely, they're going to want to put the Tremors trigger on the stack last (so it resolve first), and the opponents are dealt damage before they draw their next card (and the loop starts again). State-based actions (which is what kill players) are checked after each trigger resolves. Assuming their opponents have a lower life total then the number of cards in their library, they will kill their opponents (if one or more have a higher life total, or they stack their triggers wrong so they draw before the Tremors trigger deals damage, then they'll lose due to drawing from an empty library (since the Discovery's trigger is not optional) before they kill their opponent).
You follow the instructions in the order they're printed on the card. The artifact is destroyed first, then the creature, so the Monument is destroyed, then the Bear (which is no long indestructible).
No, Decimate is not the same. Everything is destroyed at the same time with Decimate (because it only uses the word "destroy" once), so targeting the Monument and the Bear with Decimate would mean that the Bear survives. Casualties of War uses "destroy" multiple times, so they're destroyed one at a time, not all at once.
And in limited, you can only add the main basic lands to your deck - you cannot add their snow-covered equivalents to your deck (if you want to play snow-covered lands in your draft deck, then you need to draft them). From the Magic Tournament Rules, Section 7.2 on Card Use in Limited Tournaments:
A. Mystic Compass can target any player's lands; Navigator's Compass can only target your own lands.
B. Mystic Compass doesn't let the targeted land keep its other types, so the land would lose all other abilities and just have the ability to tap for mana of the chosen color. Navigator's Compass has it keep its other types, so it would just be gaining an ability instead of only having that ability.