I don't believe Rey is Luke's daughter, for a number of reasons.
First of all, I don't see Luke Skywalker as the sort of person who'd just dump his only child on a harsh, impoverished desert world/battlefield to be "cared for" (and I use the term loosely) by a money-grubbing scavenger king who exploit and practically starves his workers. And as we know from Empire, it's not like he wouldn't be able to use the Force to check up on her from time to time. And that's not even considering that Mrs. Skywalker, whoever she might be, is just as callous and uncaring about her child's welfare.
Second, assuming that Luke's closest friends and family knew he had a daughter (which I'm assuming they would) I don't see them as those kinds of people either. Especially Leia, in my opinion, would not be pleased at the notion of her only niece wasting away in a s#$%hole like Jakku without doing something about it. Her brother's feelings on the matter be damned.
Third, judging by the flashback we saw when Rey was deposited on Jakku, she was clearly not an infant at the time. Sure, she wasn't very old, but she was walking and talking and understood that she was being left behind. Kids at that age are at the very least old enough to know their own surname, so unless we accept the premise that Mr. and Mrs. Skywalker raised their daughter to believe her last name was Antilles or something, she would have known her name was Skywalker. And there would be conversations like this one:
Rey: "Daddy, why are all the people you talk to calling you "Master Skywalker"? Isn't your name Antilles?"
Luke: "Uhh... they're just weird, sweetie. Don't mind them."
Also, the complete lack of recognition in Rey's face when she meets Luke at the end of the movie. Granted people change over time, but I had no problems recognizing Mark Hamill from RotJ all the way to TFA. At the very least there should have been a glimmer of, "Oh hey, you look an awful lot like my dad who abandonedTh me on Jakku all those years ago. What a coincidence."
Kylo Ren being Han and Leia's son if fine (spoiler alert, I guess...) but I really hope they stop at that. A character isn't only interesting because of who they're related to or descended from. Even Lucas had the sense in the prequels not to try and make Han Solo into the abandoned lovechild of Qui-Gon Jin via retcon. I'd much rather have a character who's awesome in her own right (as Rey definitely is) than because of her family name.
There are many ways that Luke could not have realized he had a child, the first one I can think of being that for some reason Rey was only raised by her mother. Which, however, then raises the question of why she would be left on Jakku. My guess is that it had something to do with Snoke, as he was awfully interested in Rey as well as Kylo Ren. (Unless, of course, she was kidnapped and sold to Unkar.)
To me there were too many pointers toward Luke in TFA for her not to be related to him.
You raise a good point with the last name question, though. I suspect that she does not know it, in fact, there's a possibility that Rey might not be her birth name. If she had known her last name wouldn't she have introduced herself with it like Poe does?
On the other hand, maybe she doesn't like to think about it as it brings back painful memories of her early childhood.

