Did Rose ever let her family know she was alive?

Pretty sure the only family she had was her mother, as the rest of the family likely shunned them once the family went broke. Marrying Cal was their only way to stay in high society, so I imagine her mother either married rich or died sooner after, or struggled in poverty. Even if she found out Rose was alive and a successful actress, if she couldn't make it to where she was living and Rose wasn't accepting letters she might've been sending her to ask for money, I don't see how she would've been able to afford to travel to her.

Of course not. She didn't want to become a "slave" again.

I think the film implies that she never reached out to them. She deliberately hid from her fiancee, and the "Goodbye Mother" line carries a feeling of finality. And when she speaks to Bill Paxton and others, the movie suggests that it's the first time she has shared her story about being on the Titanic, so she probably didn't tell anyone all those years