You pay all costs and follow all timing rules for cards played this way. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty and only if you have an available land play remaining.

The value of X is calculated only once, as the reflexive triggered ability resolves.

You don't choose a target for Cait Sith's ability at the time it triggers. Rather, a second "reflexive" ability triggers when you exile a card this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.

If the exiled card has {X} in its mana cost, X is 0 for the purpose of determining its mana value.