As for leaving him - there are a lot of sweet, committed guys out there who wouldn't cheat on you (especially not a planned cheating like you mentioned he did). And obviously it is an interest of his, not just a one time thing. He seems to be into all dimensions of it. Online, flirting, physical...It seems to be part of who he is, I think it is unlikely he will change anytime soon. Maybe in a very long time when he gets more serious.
Maybe the best thing to do would be to part with him and if it was meant to be then maybe you can try again in the future.
As for staying with him, if this is what you decide to do, then you'll have to do one of two things:
decide you're okay with staying with him even if he cheats and lies.
or
decide you want to believe he will change, and so you will need proof. ask to read his messages and snoop around. you've earned that now, he has given you no reason to trust him and so he should understand that. if you want to gain the trust back, you're going to need proof. And if he doesn't agree to let you see what he has been saying or doing, then explain this to him - you think you believe him and you just want to make sure, so you can both get back to being in love!
I know it sounds silly to ask to read his private things, but you wouldn't do it forever. Normally it would never be a good idea, but I think he owes it to you now. Denying you that would be cruel of him, given the circumstances.
And other than that you MUST get into why he cheated. Ask him. Was it the girl in particular, is he bored or unsatisfied, just a high sex drive, is he feeling doubtful about you two? Or maybe he didn't even really want to do what he did, and he was just feeling confused or maybe he was insecure about how you felt about him! Or maybe you did something to him in the past and he was feeling resentful.
He must have a reason, even if it is a very simple one, make him think about it!
I'm really sorry about how you must be feeling.

It's rough to love someone and have something like this happen...
even if you feel you should not be together, you can't just flip a switch and stop loving them instantly.
Anyway, good luck with whatever you do!