If the monarch leaves the game during another player's turn, that player becomes the monarch. If the monarch leaves the game during their turn, the next player in turn order becomes the monarch.

There are two inherent triggered abilities associated with being the monarch. These triggered abilities have no source and are controlled by the player who was the monarch at the time the abilities triggered. The full texts of these abilities are "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."

When Aragorn, King of Gondor's last ability triggers, you can choose not to target a creature, probably because you're the monarch and just want creatures to be unable to block this turn. However, if you do choose a target, and that target is illegal at the time the ability tries to resolve, the ability won't resolve and none of its effects will happen. Creatures will still be able to block this turn, even if you're the monarch.

If combat damage dealt to the monarch causes that player to lose the game, the triggered ability that causes the controller of the attacking creature to become the monarch doesn't resolve. In most cases, the controller of the attacking creature will still become the monarch as it is likely their turn.

If the triggered ability that causes the monarch to draw a card goes on the stack and a different player becomes the monarch before that ability resolves, the first player will still draw the card.

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. As a player becomes the monarch, the current monarch (if any) ceases being the monarch.