Can a countered creature be regenerated?
No, a creature cannot use Regenerate to avoid being countered.

When your friend casts Mortivore, it goes on the Stack as a Spell.  It is not a Creature until the spell resolves and leaves the stack.

Counterspell uses the Counter keyword action, which removes a spell from the stack.  This means the spell never resolves and does not have any effect.  In the case of a creature spell, the effect is to put a creature on the battlefield; if it is countered, the creature is never created.

The abilities on a creature card (including regeneration) cannot be used until the card has finished being cast and become a creature on the battlefield*.  Therefore, Regenerate cannot be used while Mortivore is on the stack.

Since Counterspell gets rid of Mortivore while it is still on the stack, your friend cannot use Mortivore's regeneration ability to avoid your Counterspell.

Note: Even if you could use a regeneration ability from the stack, it still wouldn't apply here.  Regeneration only applies when a permanent would be Destroyed, which is different than being countered.

*Except in certain exceptional cases, none of which apply here.