The second ability of Vance's Blasting Cannons counts all the spells you've cast, including Vance's Blasting Cannons itself if you cast it this turn. The ability won't trigger unless Vance's Blasting Cannons is on the battlefield as you cast your third spell of the turn.

If you don't cast the exiled card, it remains in exile.

The card exiled by the first ability of Vance's Blasting Cannons is exiled face up.

The second ability of Vance's Blasting Cannons resolves before the spell that caused it to trigger. The ability will resolve even if that spell is countered.

Casting the exiled card follows the normal rules for casting that card. You must pay its costs, and you must follow all applicable timing rules. For example, if you exile a creature card this way, you must wait until your main phase to cast it.