T1 always greets and ends with a handshake that he initiates. Although at this point I would say that it is mutual.
T2 and 3 do not say hello. They come get me in the waiting room, usually just getting my attention and then Heading back to their offices. No talk in the hallway, but I think this is both to maintain my privacy and to not be an interruption to other ts and clients. They don't really say good bye, either. Often they will say see you next week as I head for the door, but not always. To me it sounds like your t is trying to adapt for you, but doesn't get it right all the time.
You seem to have a lot of obstacles to a good therapeutic relationship with this t. I know you can't switch, and I wonder if doing what DBT calls radical acceptance would help. Basically it is accepting that things are the way that they are, even though it is not ok. For example, accepting that a family member is an alcoholic. That is not ok, but it is the way that it is. Then making decisions based on that fact, like perhaps stopping giving money, seeking support groups, not making excuses.
In your case, it sounds to me like you may have to accept that this t falls short of what you would like. For me it would be helpful to focus on the fact that t is not what I want, but is it better than nothing? Rather than focusing on what she does wrong, focus on whether I want to continue the relationship as is. It sounds like she wants to be helpful, but it also sounds like she is not going to be able to be the perfect therapist for you.