It may sounds like she is borderline, its a personality disorder. Borderlines changes from hot to cold, I think. But its for certain she is mentally ill in some way. I would also say that to throw her daughter out is abusive.

I would advice you to try say to her that things need to change and needing therapy, maybe not just for her but for you too. Hanging onto unhealthy relationships are something we do when we have FOO issues. Some may not have FOO issues, but many has. I have FOO issues and I held onto abusive relationship because I thought the man would change. It never happened because they were sociopathic/narcissistic/abusive.
Borderlines I dont know if they can change, I guess borderlines have genuine feelings but they cant control the hot and cold. I dont know.

I would advice you to tell her things must change and keep yourself out of the relationship until you see a change, if that ever will happen. It may happen only if she agrees.

You must be very hurt by how she treats you, breaks up with you, keep you hanging, its not fair you be treated like this. Abusive relationships are damaging to your soul and well being and self esteem.

I know how hard it is to get out from an abusive relationship, but I do hope you will take all the good advice you have gotten from everyone here and take care of yourself and your son. I do hope this womans daughter will get a safe home, because right now it seems like child abuse.