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.

The instants or sorceries cast using copies of Storm of Memories go on the stack as each copy of Storm of Memories resolves, and those spells will each resolve before more copies of Storm of Memories resolve.

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.

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.

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.