No, not if your mother has sole physical custody. If there is a joint custody arrangement, you'd still need the cooperation of both parents to make it happen.
Only if you are 18
no
Only with their permission. The custody order have to be changed in the court it was issued.
30 miles
No.
It depends how old she is and she should at least let him know
Definitely,decisively,with a strong emphasis on the part 'without your mothers'.Mine fathers don't live, it is a reason and causae of mine version.
No you can legally move out when you are 18 no matter what state you are in.
Not unless you were married (parental approval), or by court order.
In most places, you can move out of your mother's house legally if you are at least the age of majority (usually 18) or have been emancipated by a court. If you are under the age of majority, you may need to get her permission or involve the legal system to establish your ability to live independently. It's best to research the laws in your specific area to understand your rights and responsibilities.
That is dependent of custody orders, but even without, the other parent could file an injunction.
beyond 50 miles