Does a copy of a spell go on the stack and can it be countered?

When an effect creates a copy of a spell, that copy is placed on the stack as a new object (CR 706.10). It exists independently of the original spell and follows all the same rules that any spell on the stack follows.

Because the copy is on the stack, players receive priority and may respond to it. Any spell or ability that can counter a spell — such as Counterspell — can target and counter the copy (CR 701.5a). Countering the copy does not affect the original spell, and countering the original does not affect the copy.

One important nuance: some copy effects specify that the copy 'isn't cast.' In that case the copy is still placed on the stack, but abilities that trigger 'when a spell is cast' will not trigger for it, since casting and placing a copy on the stack are distinct actions (CR 706.10).

Concrete example: Your opponent casts Fork, copying your Lightning Bolt. The copy of Lightning Bolt goes on the stack above the original. You respond by casting Counterspell targeting the copy. The copy is countered and removed from the stack; the original Lightning Bolt is unaffected and resolves normally.