If you cast a spell with in its cost without paying its mana cost, you must choose 0 for X.

The spell that caused Arcane Bombardment's ability to trigger is normally still on the stack while that ability resolves. That spell won't be among the cards in your graveyard when you exile an instant or sorcery from it at random.

If you've already cast an instant or sorcery spell the turn that Arcane Bombardment enters the battlefield, it won't trigger until you cast an instant or sorcery spell on a later turn.

If Arcane Bombardment leaves the battlefield, the exiled cards remain in exile. If it returns to the battlefield or you play another copy of it, you won't get to cast copies of the cards previously exiled.

You cast the desired copies as the triggered ability resolves. You don't get to cast them later in the turn. You choose the order of copies as you cast them.