If you cast a spell "without paying its mana cost," you can't pay any alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, you must pay those.

You cast the copy while the ability is resolving and still on the stack. You can't wait to cast it later in the turn.

The three cards you exile are chosen at random from among all cards in your graveyard. If all three cards you exile this way are creature and/or land cards, you won't be able to choose one of them to copy and cast. Capricious, indeed.

If you don't want to cast the copy, you can choose not to; the copy ceases to exist the next time state-based actions are checked.

If the spell you cast has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.