To copy a spell, activated ability, or triggered ability means to put a copy of it onto the stack; a copy of a spell isn't cast and a copy of an activated ability isn't activated. ... A copy of a spell is owned by the player under whose control it was put on the stack.

Is copying a spell playing it?

Nope. Copies are put onto the stack, not played.

Do copied spells count for Storm?

The storm ability only counts spells that are cast. It does not count when copies of spells are put on the stack.