The copies are put directly onto the stack. They aren't cast and won't be counted by other spells with storm cast later in the turn.

Spells cast from zones other than a player's hand and spells that were countered or otherwise failed to resolve are counted by the storm ability.

A copy of a spell can be countered like any other spell, but it must be countered individually. Countering a spell with storm won't affect the copies.

The triggered ability that creates the copies can itself be countered by anything that can counter a triggered ability. If it is countered, no copies will be put onto the stack.

If a spell with storm has targets, you may choose new targets for any of the copies. You can make different choices for each copy.