Does the card (support or unit) that has an action which can be triggered have to be in play at time of resolution?

For example, if my opponent stacks 5 actions of using Repeater Bolt Thrower (with 1 development in the battlefield for 1 indirect damage per action) and I pillage the Bolt Thrower before these actions get resolved, do these actions still resolve even though the Bolt Thrower is destroyed?

Also, what if the development gets destroyed? Does X then become 0? If we follow the ruling of Deathmaster which states you need to have enough skaven for the action to resolve both at time of activation and at time of resolution then it makes me wonder...

Any thoughts or answers?

Yes. Previous actions would still resolve as they are on the stack already. However, destroying the development would cause the damage to decrease by 1. Hence 1 indirect becomes 0 and so on.

Sining, you are correct on part 1 but incorrect on part 2.

The X in the effect is fixed at declaration time because you have to pay the cost then. Getting rid of the developments after that doesn't change anything on the stack.

Killing the Thrower as per usual doesn't do anything to anything on the stack because removing the source of the effect doesn't remove the effect.

Okay. Would that mean that if a player decides to use lokhir fellheart to kill off a unit with his special ability, destroying a development on top of that stack would do nothing to prevent it?

No, Lokhir works the same way as Innovation. If you destroy a development in response to his Action, then he won't give the target as many -HP. The Thrower is different because you have to pay a cost with X in it (which fixes the variable X value).