Players can cast instant spells (including instant spells that would deal damage to you) in response to Aetherflux Reservoir's last ability as they can to any other activated ability that goes on the stack (C.R. 117.7, 117.1a). Like any other activated ability, Aetherflux Reservoir's last ability has no effect unless and until it resolves (C.R. 608.2c, 609.1). And Aetherflux Reservoir's last ability can resolve only if all players pass in a row while that ability is on top of the stack (C.R. 117.4, 608.2c).
EDIT (Jan. 1): Add rule citation.
EDIT (Dec. 8, 2019): Some rules were renumbered with Core Set 2020.
So I'm more than a little bit fuzzy on priority and passing between players.
I have 4 opponents.
I activate Aetherflux Reservoir's second ability, then Player one gets a chance to respond to the activation, then player 2, then player 3, then player 4 and if no responses the ability resolves.
Is that correct?
Also when do I have to declare a target? When it is first activated and before everyone gets a chance to respond or after they have a chance to respond and it begins to resolve?
Yes, you have it correct. Note that you paid the cost of activating the ability already so while priority is passing, you are at 4 life.
You declare your target when you put the ability on the stack (this is the same for all spells and abilities). So, each person knows who is targeted when they get priority.
I am at 54 life.
I activate the second ability on the reservoir and target my opponent for 50 damage.
Can they cast spells to kill me before they die?
EDIT (Jan. 1): Add rule citation.
EDIT (Dec. 8, 2019): Some rules were renumbered with Core Set 2020.
I have 4 opponents.
I activate Aetherflux Reservoir's second ability, then Player one gets a chance to respond to the activation, then player 2, then player 3, then player 4 and if no responses the ability resolves.
Is that correct?
Also when do I have to declare a target? When it is first activated and before everyone gets a chance to respond or after they have a chance to respond and it begins to resolve?
You declare your target when you put the ability on the stack (this is the same for all spells and abilities). So, each person knows who is targeted when they get priority.