The game starts with no monarch. Once an effect makes one player the monarch, the game will have exactly one monarch from that point forward.

Being the monarch carries two inherent triggered abilities. "At the beginning of the monarch's end step, that player draws a card" and "Whenever a creature deals combat damage to the monarch, its controller becomes the monarch."

If you're not the monarch as Palace Jailer's 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.

Auras attached to the exiled creature will be put into their owners' graveyards. Equipment attached to the exiled creature will become unattached and remain on the battlefield. Counters on the exiled creature will cease to exist.

If a creature token is exiled, it ceases to exist. It won't return to the battlefield.

Palace Jailer leaving the battlefield won't cause the exiled creature to return. The game will continue to watch for the next time an opponent becomes the monarch.

The opponent that controlled the exiled card doesn't have to be the same opponent that becomes the monarch in order to cause that card to return to the battlefield. Any opponent becoming the monarch will cause the card to return.

In a multiplayer game, if the monarch leaves the game, the player whose turn it is immediately becomes the monarch. If the player whose turn it is has left the game, instead the next player in turn order becomes the monarch.