Rei is my favorite guardian soldier in the manga and live action series. I love that she's very independent and a bit feisty while maintaining a sort of cool demeanor. The fact that she has no interest in men is one of the things that I like about her character. I don't like that she thinks all men are bad, but I think that is probably something that would change with maturity. I wouldn't mind if she eventually had a love interest, and yes, I do think that could eventually happen for her. It's just that Usagi, Minako, and Makoto are so boy crazy that it's refreshing to have characters (Rei and Ami) that have different priorities.
She couldn't be lying about having psychic powers, because specific situations that she predicts eventually come to pass. Her strong spiritual abilities is one of the most defining aspects of the character. There is nothing to indicate that she is lying, so it is pointless to suspect otherwise.
I don't have a strong opinion about her parents - of course, her relationship with her father is sad, and the audience is meant to sympathize with Rei. I like her grandfather, but he really doesn't have enough character development to be very interesting to me.
I don't like what the anime did with Rei because I'm very fond of her character in the manga. I don't think it was necessary to change her character so drastically. In the manga, she can be a bit temperamental, but usually she's able to keep her cool. The anime takes her temper to the extreme, and makes her relationship with Usagi much more combative. For whatever reason, the writers also felt compelled to change her dream, interests, and talents - in my opinion, this combined with the changes to her personality make her a very different character. I'm very much a fan of Rei as she appears in the manga and live action. In the anime... not so much.