There can be issues associated with same sex relationships, and many of those revolve around the degrees we live in the closet. This is really a case though where it really does not matter if its a same sex relationship or not. Break ups are well heart breaking, what makes things worse is that neither of you have the courage to make a clean break. I guess since your gf dumped you, she should have been the one to make the clean break. What you are doing is bargaining, seeing what you can salvage out of the relationship. "Lets be friends" is the worst thing you can do right after a break up.
Cut off all contact, forever, have a ceremonial burn of all her stuff, hate her, let her go.