I am SO sorry you're going through this trauma. I, too, was betrayed by a man, who was unveiled to be very different than I thought he was. I understand the crushing disappointment and depression. The very first thing you need to do right now is take care of yourself, and your emotional well-being. If you can find a counselor ASAP, that would be a great first step.
For me, this situation would result in an immediate dissolution of the relationship. It is possible that, through individual and marital counseling, that the relationship could be improved, but I personally feel that trying to repair this situation will only prolong the inevitable, and prolong your pain.
Do you have any family or friends who could help you?
I'd take hope in the fact that YOU didn't do these things, HE did these things. If you did anything wrong, it was to pick the wrong guy, believing he was a decent guy. There is nothing wrong with believing. You probably got some lift from the early days of the relationship, too, because it gave you hope that you had an ally in fighting all your demons. Think of that trust you had, and that temporary lift the relationship gave you, as good and honorable things. You are not at fault, you are not worthless. (He might be worthless, but that is another story.) You are a good person who believed. Betrayal is very sad, and we all grieve for someone whose trust has been abused. But YOU did NOTHING WRONG. All you did was trust.
If you don't have family or friends to help you, do you have any of the hotline numbers? They can be really helpful, and are not judgemental.
I have a bit of a different take on it. Two people with mental issues get together and hope they will be good for one another. Might happen. Might not. But solving your mental problems or him solving his is unfortunately probably not something you're going to achieve or he's going to achieve from a romantic relationship. That help comes through hope, work, time, therapy, and all that stuff both of you have undoubtedly been trying. It's hard. I have no opinion on the fact he likes sex the way he likes it on the internet. A lot of people are doing that. Maybe most people these days, who knows? I wouldn't want my partner doing that, but say I didn't know, the best judge of a romantic partner is how they treat you, not their secret sexual fetishes, which we all have. The relationship with the sister, on the other hand, if it is actually ongoing has to be a deal breaker. It's something he hasn't fixed and is so far out of bounds and I can't see how it can be easily or quickly fixed if it hasn't been fixed by now. I say assuming it is ongoing, because you aren't clear on that. People who have been sexually abused do a lot of very troublesome things. Coping is hard. That alone can't be a deal breaker if we are to be kind and loving humans. That doesn't mean, though, that a person is ready for a relationship. But if it's ongoing, he's not only cheating on you, he's cheating on you with his sister. I don't think you can fix that especially given you have your own troubles to deal with. I think in this situation self-protection outweighs all else. Peace.