Hi,
Your first situation is helped out by this Gatherer ruling:
If you’re not the monarch as the second ability resolves, the creature will be exiled until there’s a new monarch and that player is one of your opponents. The creature won’t immediately return just because an opponent is the monarch.
It does not matter what happens with the monarch while the Jailer's exile trigger is on the stack, so changing who the monarch is with the trigger on the stack has no effect on the trigger. Once the Jailer's triggered ability resolves, the exiled creature will not return until one of your opponents becomes the monarch, and that's the point when the effect will care about an opponent becoming the monarch.
As for your second question, it is also covered in the Gatherer rulings:
Abilities that trigger whenever you “become the monarch” trigger only if you aren’t already the monarch. For example, if you are already the monarch as Custodi Lich enters the battlefield, its last ability won’t trigger.
While not a trigger, it still covers this situation - in order to become the monarch, you have to go from “not being the monarch” to “being the monarch”. Becoming the monarch when you're already the monarch won't do anything (and will not cause the exiled creature to return).
Nathan Long
Magic Rules Guru