From page 14 of the rules reference
b. Choose Weapons: The attacker can choose one weapon from each eligible mini to contribute to the attack
Therefore it seems you could choose the regular squad weapon if the special weapon is exhausted.
I think so. Even if exhausted, the miniature is still on the ground then he could attack with his Unit. Sure it would be nice if someone wrote it in the Rules Reference, maybe in the Exhaust topic (pag. 27)