Did Rose ever let her family know she was alive?

Maybe she updated her Facebook status.

Doubtful. She wanted nothing to do with them or their lifestyle. As far as Rose was concerned, this was a completely clean slate.

Getting in contact with them would mean a chance of them getting a grasp of her. They were all dead to her anyway. She despised all of them.
The "Goodbye Mother" was most definately meant as just that.

And dont forget, their golden ticket literally tried to murder her.

It was implied she did not and even took Jack's last name hiding her identity. She started over in America.

Likely no, but if she became an actress as it was said of her, her family would have recognized her, so likely a writing screwup.

NOT sure how much family she had aside from her mother

AND without the meal-ticket future son in law where was she going