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

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

You choose which spells (if any) to cast as Villainous Wealth resolves. If you do, you do so as part of the resolution of Villainous Wealth. You can't wait to cast them later in the turn.

Any cards you don't cast this way will remain in exile.

You cast the spells one at a time as Villainous Wealth is resolving, choosing modes, targets, and so on. The last spell you cast will be the first one to resolve. Ignore timing restrictions based on the spells' types. Other timing restrictions, such as "Cast [this spell] only during combat," must be followed.