Given that his x-ray vision allows him to see though skin, can he use this ability with his eyes closed? Or do they prevent it somehow?
How Superman X-ray vision works varies from comic to comic, but there is a feat that could prove that indeed it is possible for Superman to see through his eyelids: a Kryptonian seeing through his own skin.
Below you can see a picture (Supergirl #2 - actually I got the it from a Google images search that linked this question, so thanks) that shows Supergirl being able to see her own bones through her skin.
Would this cause Superman to be unable to sleep? Thankfully for him it's been shown repeatedly that he needs to purposely activate his power.