You pay all costs and follow all normal timing rules for cards played due to the last ability. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

You don't choose a target for Inti, Seneschal of the Sun's first ability at the time it triggers. Rather, a second "reflexive" ability triggers when you discard a card this way. You choose targets for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.