Is it a problem if eyelids don't fully close during sleep?

Maybe: If your eyes are irritated and/or red, then this is a problem. If not, don't worry about it. Many people tolerate incomplete eyelid closure well. If you are concerned, see an ophthalmologist who can examine and advise you.

Eyes not closed: I recommend speedo swimmer's goggles. It seals the area and prevents evaporation of the remaining tears. It has an elastic band that does not interfere with sleep, or tend to come off during sleep.