He seems interested, but is he single? It's confusing to tell?

[JTM]

Ok, so I have never had anything like this happen before, and honestly, it's a little confusing. There is this guy at my college who I share classes with. I have noticed that about halfway through this semester he began initiating conversation with me and seems as though he might be interested in me.

He goes out of his way to make eye contact and say Hi! happily across a room full of students trying to quietly study.
He consistently asks me for "help" with homework, even though I have (embarrassingly) made it pretty obvious that I am no better at it than he is. He does this even when there is a professor present as well as when there are other students at this table that he talks to frequently (he will leave them and come to me instead).
Also, he never asks a direct question pertaining to HW that "makes sense," it's always more of a general conversation starter.
He also smiles, blushes, and laughs lightly a lot when we talk, and his eyes just seem "happy," which is rare for him - he's usually a serious, straight faced guy with others.
I have also caught him watching me, and he full-body faces me whenever we are together.

The thing is, on his FB page there is a picture of he and a girl from two years ago. I went to the girl's page, and there is a single pic of them together from this Feb - but that's all; no convos between them, and they don't even "like" each others pics. They are pretty FB active otherwise.

Any thoughts, because I'm confused?
Basically, I am wondering two things - am I correct in thinking that he likes me (from the vague description that I gave there), and do you think he is dating this girl?

The odd part is that he acts as if he is very into me - I've never been treated this way by someone that wants to be "just friends," only by guys that want more. He also seems to be completely single (as does she) - his social media suggests so other than those two pictures.
I think the main reason I am asking this question, too, is because I was literally about to make the first official move. He seems hesitant, so I was going to casually suggest that we "study together" sometime - and give him my email / number / etc. I just feel uber awkward doing that if he is already with someone. Do you think that I should still do this? I am very into him...just...hesitant now...

Sorry for all the "updates," but I couldn't fit this all into the main body of the question.

[Guadalupe]

Just talk to him

[Andrew]

He defintley likes you,as it seems he dosen't do it for everyone. As for the girl,they could be cousins,or childhood friends,explaining no real facebook presence outside of the pictures

Answer mine?https://answers.yahoo.com/question/index?qid=20140404205614AAtud77

[Holly]

AWWW that's so cute. He clearly thinks you're pretty, and I bet they were just old friends or ex's. You should ask him on a casual date, like coffee or something because I bet he was going to do that anyway.