I think you should give her the benefit of the doubt... it sounds like she does really like you (after all, you seem certain she loves you) and if this is true then why are you worrying about the other 2 things? It's not as if she's unreliable, cancelling on you for going out with her friends. To me, it sounds like she has a rather protective family and she's worried that if she tells them about you, they won't let her out anymore. I had a friend whose family was like that and it was so humiliating and mortifying for her, she just wanted to be a different, stronger, independent girl for her boyfriend so of course he didn't know, she wanted to be the best for him. Have you tried talking to her about how you actually feel? And you need to consider that she might actually be in a tricky situation so don't let the fact that you're pissed off get in the way.
Don't forget that parents for guys and for girls are completely different things. As a girl, I know that my mum was always super protective when it came to boyfriends; she literally could not relax! And saying "if she knew about us, she might understand and let her stay at mine..." no no no, definitely not true! I had to lie to my mum when I stayed at my boyfriends! I had to tell her that either one of our mutual, female friends was there or that it was a big sleepover, we were staying in different rooms, yes his parents were always checking up on us...
Girls parents are so much more protective. And you just need to show her that you are trustworthy. Also, 4 months isn't that long, you have plenty of time ahead for her to tell her parents and for you both to open up to each other. Which you kind of need to do.