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."

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.

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.

The last ability of Keeper of Keys will affect all creatures you control that turn, even if they weren't on the battlefield or weren't creatures as the ability resolved.

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 last ability of Keeper of Keys checks to see if you're the monarch as your upkeep begins. If you're not, the ability won't trigger at all. You won't be able to do anything that would make you the monarch during your upkeep in time to have that ability trigger. The ability will also check to see if you're the monarch as it tries to resolve. If you're not the monarch at that time, the ability will have no effect.