Usually after closing my eyes and trying to get to sleep for two or three minutes, my eyes seemingly fuse to my eyelids and it is uncomfortable moving them around. I open them up (which is also uncomfortable), blink a few times, and close them again and the problem goes away for the night.
Are my eyes dried out or something?
I wouldn’t know for sure, but it sounds like your eyes just might be dried out. Other than that, I have no idea what could possibly be happening to you. 
You should probably talk to an eye specialist about that, see about getting an ointment of some sort to moisten the eyes.
If it’s that bad it could cause problems with the cornea, particularly through REM sleep.
Sounds like conjunctivitis to me. I woke up one morning and just had this problem out of nowhere - it kept persisting for weeks. When I had an eye appointment they found it was conjunctivitis. Apparently your eyelids stick to your eyes because the glands there are secreting this sticky gunk which dries out if you don’t blink (thus getting stuck to your eyelids). So I would definitely recommend getting this checked out at a doctor, because conjunctivitis only gets worse if it’s not treated.
Well it only happens once during the night, though I think I may get it on occasion when I wake up. I don’t remember for sure. I’ll be observant.
My eyes. MY EYES!
EDIT: I just looked up conjunctivitis on Wikipedia, and I don’t seem to fit the symptoms. My eyes aren’t really red, though they are a bit more veiny than usual because my sleep schedule, frankly, has SUCKED this semester. I don’t get any goop or the like in my eyes, and my eyes do not itch. They just get irritated after closing them for three minutes or so, but it goes away after blinking. I never get it more than once per night.