You pay the costs for an exiled card if you cast it. You may pay any alternative costs the card has rather than the card's mana cost.

The six types of mana are white, blue, black, red, green, and colorless. Snow mana is not a type of mana. Court of Locthwain won't let you pay a snow cost using mana produced by a nonsnow source.

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.

You may play the exiled cards (and spend mana of any type to do so) even if Court of Locthwain leaves the battlefield. If another player gains control of Court of Locthwain, that player can't play the cards, and you still can.

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."

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 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.

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.