Dies to Removal

Conjurer's Closet
Zealous Conscripts

So, three scenarios, each starting with Conjurer's Closet, Zealous Conscripts and the permanent (let's just say it was a Grizzly Bears) that it stole this turn:

1. Beginning of end step, place Conscripts trigger on the stack first, then Closet trigger. Closet targets Bears, resolves. Do I get to keep the Bears?
2. Beginning of end step, place Conscripts trigger on the stack first, then Closet trigger. Closet targets Conscripts, resolves, Conscripts ETB, targets an opponent's Doubling Season. Do I get to keep the Bears? Do I get to keep the Doubling Season?
3. Beginning of end step, Closet trigger on stack first, they got their Bears back already. Bounce Conscripts, steal Bears. Do I get to keep Bears?

Is there even a trigger associated with the "gain control until end of
turn"? If so, with what permanent is it associated--the Conscripts or the stolen one?

There is no Conscripts trigger associated with the control changing back. Control reverts to your opponent at the end of turn, during the cleanup step.

Closet triggers at the beginning of the end step, so you're free to blink any creature you control and keep it permanently before the control granted by Conscripts wears off.

You're assuming there's a trigger for your Conscripts, so your questions are a bit off. I'll answer them, sort of.

1. Steal bears, at EOT, target beats with closet, you keep them.

2. You can't target the Doubling Season with Closet, and double blinking the Conscripts won't let you keep it.

3. If you blink Conscripts with closet EOT you'll gain control/untap the bears for a second time, but you'll lose it during cleanup step.