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.

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.

The game starts with no monarch. As a player becomes the monarch, the current monarch (if any) ceases being the monarch. There is never more than one monarch at a time.

If some of the creatures are illegal targets as Court of Garenbrig's last ability tries to resolve, the original distribution of counters still applies and the counters that would have been put on illegal targets are lost.

To double the number of +1/+1 counters on a permanent, put a number of +1/+1 counters on it equal to the number it already has. Other cards that interact with putting counters on it will interact with this effect accordingly.

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."

You choose how the counters will be distributed as you put Court of Garenbrig's last ability on the stack. Each target must receive at least one +1/+1 counter.