Hi Tut,
Welcome to the group

She is starting to show bpd behaviors towards my daughter, e.g. claiming that my daughter should do things for her since she invested in her so much until now, or talking about how I'm making my daughter miserable since her siblings were born. She is like obsessed with her.
This sounds like FOG (Fear,
Obligation,
Guilt) or emotional blackmail. When there is pressure to do something you don't want to do it is usually FOG. It helps when you can recognize the FOG for what it is.
More on FOG...
https://bpdfamily.com/message_board/index.php?topic=82926.0When she is in a bad situation I don't let her see my daughter for a while and she is saying that I'm distancing them and that she'll go to the court to have visiting hours.
Good for you for protecting your daughter, you are setting an appropriate boundary here. This is about protecting your daughter and not punishing your mother.
I'm terrified of what will be in the near and distance future. E.g. when my daughter will have a phone and my mom can have a direct access to her
I swear cell phones are the bane of our existence...my partner also had trouble with his undiagnosed BPD ex-wife and their children's cell phone. She had no qualms about calling her daughters and waking them up in the middle of the night on a school night.
You will likely want to talk with your daughter about rules around a cell phone once you get her one. We also have a co-parenting board that you might find helpful when the time comes to get your daughter a cell phone or other issues regarding your mom and daugher...
https://bpdfamily.com/message_board/index.php?board=10.0 I'm curious what you think your daughter's awareness is regarding the issues your mom has and the difficulties you and your mom have?
Panda39