The ability triggers when a player casts an instant or sorcery *card*. A copy of a card (such as those generated by Eye of the Storm itself) won't trigger this ability. The card that was cast is exiled and won't resolve, but the player will get a copy of the spell from Eye of the Storm.

When the ability resolves, the player copies all cards exiled with Eye of the Storm, not just the cards he or she owns. That player then chooses which of those copies to cast, if any. All chosen copies are cast in the order the player chooses.

The player chooses modes, pays additional costs, chooses targets, and so on for the copies when casting them. Any X in the mana cost of a copy cast this way will be 0. Alternative costs can't be paid.

You must still follow any restrictions on when you can cast the spell, such as "cast only during combat" or "cast only on your own turn."

When a spell with Replicate is copied by Eye of the Storm, you have the opportunity to pay additional costs, so you may Replicate as much as you like (assuming you have the mana available).