When Court of Vantress becomes a copy of a permanent, it's neither entering nor leaving the battlefield. Any enters-the-battlefield or leaves-the-battlefield abilities won't trigger.

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.

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."

Court of Vantress copies the printed values of the enchantment or artifact plus any copy effects that have been applied to it.

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 the copied permanent is copying something else, then the token enters the battlefield as whatever that permanent copied, with the stated exceptions.

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.

Any enters-the-battlefield abilities of the copied permanent will trigger when the token enters the battlefield. Any "as [this permanent] enters the battlefield" or "[this permanent] enters the battlefield with" abilities of the permanent will also work.

The copy effect lasts indefinitely. Often, it will last until it is overwritten by another copy effect (if it copies another permanent on a future turn, perhaps).

If the copied permanent has in its mana cost, X is 0.

If Court of Vantress becomes a copy of a token, it copies the original characteristics of that token as defined by the effect that put it onto the battlefield. It won't become a token.

If another permanent becomes a copy of Court of Vantress, it will become a copy of whatever Court of Vantress is currently copying (if anything) that also has Court of Vantress's last ability.

If Court of Vantress becomes a copy of an Aura, it's put into its owner's graveyard unless it's somehow attached to an appropriate object or player already. If it becomes a copy of an Equipment and is attached to a creature, it'll become unattached when it becomes a non-Equipment, non-Aura permanent.

The token copies exactly what was printed on the original permanent and nothing else (unless that permanent is itself copying something else; see below). It doesn't copy whether that permanent is tapped or untapped, whether it has any counters on it or Auras attached to it, or any non-copy effects that have changed its power, toughness, types, color, and so on.