Yes they still have the critical injury after they have recovered. The same as if someone would use the Heal skill to treat a broken character or a some healing drug. They heal the trauma, not the critical injury.
I don't think it is written anywhere what weapon is two handed or not. Although it is fairly obvious. Personally, I treat any heavy weapon as a two handed weapon but also spears, rifles and bows.
“I’d say thistles, but nobody listens to me, anyway.” - Eeyore