You will always get priority back first when you cast a spell or activate an ability. If you don't make it clear that you're going to maintain priority to cast another spell/activate an ability then it's assumed that you have passed priority. So because you've passed priority, if you're opponent then passes priority too, the topmost item on the stack will resolve before anyone receives priority again.
Short version: No you can't activate the vial in this scenario.
The Mosquito has a standard triggered ability, which starts with "When...". The ability triggers when the Mosquito enters the battlefield, which it says right on the card. The ability will use the stack, and you can respond to it with the Artillery activated ability. The Mosquito will take 2 damage and you will take 3 damage, then the Mosquito will die. Then your Artillery will die from the Mosquito's ability.
Yes. The tokens that die will go to the graveyard, it's just that they won't stay there very long. They will cease to exist the next time state based actions are checked. Anything that triggers from a creature dying will trigger.
The way that blocking works is that the attacker chooses an order for the blocking creatures, and then must assign at least lethal damage to each one before assigning damage to the next one. So in this case he can assign anywhere from 1-9 damage to the first blood artist, and then the rest to the second one. If he chooses to assign 9 to number one, then number two will have 0 assigned and will live.
Firstly, Devour isn't a triggered ability, it's a replacement effect.
Secondly, you choose which creatures to devour all at one time, which is when the spell is resolving. The Thrinax has a triggered ability that triggers when it dies, so you wouldn't be able to devour the tokens from the Thrinax.
Here's how it would play out:
You cast Mycoloth and pass priority.
Opponent(s) pass priority.
Mycoloth begins to resolve.
You choose the elemental tokens and the Thrinax to sacrifice to Devour, and the creatures are put into the graveyard.
Thrinax triggers, but the ability isn't put onto the stack yet.
Mycoloth finishes resolving.
Thrinax ability is put onto the stack.
You pass priority.
Opponent(s) pass priority.
Thrinax trigger resolves and you get 3 tokens.
Yes. The Helix will resolve before the Doom Blade. And then both players will get priority again before the Doom Blade resolves, at which point you can activate the ability granted by the helix.
Nothing would happen, the game would continue. You would try to draw a card, Lab Maniac would try to make you win but Platinum Angel prevents it. In Magic, "can't" always wins out over "can".
Unfortunately not. Since Overgrowth's triggered ability is a mana ability it doesn't use the stack. And the Resonator needs to be able to target the ability to copy while it's on the stack.
Anything in the game that changes zones is treated as a completely new object, with no memory of any previous existence. In this case it will be an untapped creature with summoning sickness.
Short version: No you can't activate the vial in this scenario.
http://www.wizards.com/Magic/Magazine/Article.aspx?x=grandprix/welcome
Firstly, Devour isn't a triggered ability, it's a replacement effect.
Secondly, you choose which creatures to devour all at one time, which is when the spell is resolving. The Thrinax has a triggered ability that triggers when it dies, so you wouldn't be able to devour the tokens from the Thrinax.
Here's how it would play out:
You cast Mycoloth and pass priority.
Opponent(s) pass priority.
Mycoloth begins to resolve.
You choose the elemental tokens and the Thrinax to sacrifice to Devour, and the creatures are put into the graveyard.
Thrinax triggers, but the ability isn't put onto the stack yet.
Mycoloth finishes resolving.
Thrinax ability is put onto the stack.
You pass priority.
Opponent(s) pass priority.
Thrinax trigger resolves and you get 3 tokens.
Lorthos, the Tidemaker
Overburden can also serve a similar role against a lot of decks.