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.

Protector of the Crown's last ability doesn't depend on you being the monarch. It will create a redirection effect even if an opponent is the monarch.

Applying this redirection effect doesn't change whether the damage is combat damage.

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."

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.

If you control more than one Protector of the Crown, you choose which redirection effect to apply. You can't divide damage dealt by one source. For example, if an attacking creature would deal 6 damage to you and you control two Protectors of the Crown, you may have that damage dealt to either of the Protectors. You can't have 3 damage dealt to each one.

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.