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I can't say for sure whether your girlfriend loves you or not, but based on what you've written about your relationship it sounds like you don't really FEEL loved. If you're not getting much physical affection, don't feel appreciated, and worry that she's not trustworthy, it sounds like the relationship could be very stressful and unfulfilling for you. My guess is that there are things about the relationship that you like or you wouldn't have stayed in it for so long, but I wonder if those things outweigh these painful aspects that you have described here.
I would suggest that, rather than focus on whether or not she loves you, you focus on whether or not this relationship is making you happy. If so, then great. If not, what would you need to change to make it so that it's fulfilling for you? What would you need her to change? It would be optimal if you and she (together) are willing to work on those things.
I don't know what causes your girlfriend's rages- fits of rage can be caused by lots of things including (but not limited to): bipolar disorder, extreme anxiety, hormonal imbalances, post-traumatic stress disorder, sleep deprivation, etc. It sounds as though your girlfriend could use some professional help in sorting out why she is so angry with you and has so much trouble accepting responsibility for working on issues in your relationship. Unfortunately, you cannot force her to get professional help, nor can you force her to listen to you. Again, it might be more useful to think about how you'd like to handle your distress over this rather than focusing on how to change her.