Thank you for that information, it was helpful. You are dealing with different scenarios here, and I'll address them individually and then talk about how they are related.
The cross dressing is a concern because I'm not sure if he's being totally honest about it. It's strange to me that he would just out of the blue decide to purchase breasts with no prior indication, and that you would accept it. Most women would have freaked out, maybe even left, but you didn't? Perhaps you suspected something already? By allowing him to make these purchases and wear them around you, you sent him the message that you are willing to accept it. There's nothing wrong with accepting it if you are actually accepting it, the problem is, if you accept it, then you can't accept it sometimes and not other times. Cross dressing doesn't mean he's gay, but it still requires a level of acceptance from you.
As for this other woman, he has no right to become angry at you, you were not the one who cheated. If anything, he should be trying to prove to you that he made a mistake and that he wants to make things right. He is the one that lied and cheated, he knows this, I agree that his anger may have a lot more to do with getting caught than anything else.
At this point, he needs to tell you how you can trust him from this point? How do you know he won't cheat again? Even though he claims he did not sleep with her, he put himself on a dating site although he's been in a relationship, that is still cheating. By having this other involvement with this woman, he was lying to you every day, and being deceptive.You're the one who deserves to be angry.
He's been lying about the cross dressing since you met, and he lied about this woman and having a singles profile. It makes you wonder what other things he lied about. The first thing is to talk to him about that, and ask him if he has anything else he needs to tell you and how you are supposed to trust him in order to move on? How are you supposed to buy a house with him, and commit to him if he's not willing to be honest? I think that counseling or therapy is something that he would benefit from, in fact couples counseling should be a prerequisite to the two of you buying a house, so that you can try and get everything out in the open and deal with everything together.
First you have to decide if you want to be with a man who cross dresses. There are many women who are very happy with their husbands or boyfriends who cross dress, so it's not unthinkable that you could be happy with him and learn to love this part of him. You will need to talk to him and find out what he gets out of it.....does he want to wear it in the house? out of the house? around other people? in bed? during sex? Then you have to decide if you are comfortable with whatever his needs are. If you don't want him talking to other women and having a singles ad, you have to be clear about that and let him know what you want, and he has to decide if he can comply. Relationships are about love, trust and compromise.....you can get through this if you want, but those three needs still need to be met above all.