So I want to join an LGBT support group on Facebook. But I'm not open with my sexuality and I don't want anybody that I don't want to know to know. So would my FB friends see that I'm in an LGBT Facebook group? Will they see my posts?
Yes. Facebook privacy "settings" are extremely complex and carefully designed to screw you over. The complex interactions that occur with "shared" items that have different audience settings is difficult to figure out, and change frequently. Also, sometimes comments on shared items can be public depending on who is sharing them, which means that people you know can use "metadata" to reverse engineer. You could consider using a fake name and another account, not in any way linked to you. An alias in other words. Just remember that pictures can be "tagged."
To be honest, Facebook is one of the sites I trust the least. As Faustian said, I wouldn't be surprised to get screwed over. The only thing I can think of are the "Secret groups" but I don't think you'd be able to find them...