Shh - yes, called ID is fantastic. I have visual and talking caller ID on my home phone so I can hear who is calling and I love it. No racing for the phone if it's not who I want to talk to. This lets me choose when I talk to family and friends. I have it only mobile too.
It sounds like your mom is really testing the boundaries of your new relationship with her. I wonder if some of it is fear of being far away from you and is an escalation that will equalize after she moves? It does sound very frustrating. However, you sound very strong and sure of yourself.
Keep following your instincts. I think they are right on.