If you live in the US and girlfriend is a minor, her parents don't need any other reason other than 'I don't want him around my daughter'
Yes they can.
probably
Yes. Your g/f's mom is her legal guardian and can get a restraining order against your regardless of your age. If you violate the restraining order both you and you parents will be legally liable.
Try to prove and show them that you are a nice, kind, caring individual. If they believe that you are a good man for their daughter, age shouldn't matter.
Kick their door in, and invade their privacy. Yeah then they can call the cops and get you arrested.
They can take steps to protect their child. Yes, the child is their responsibility.
deal with it - everything has consequences
Yes, they can. Parents have the right to determine who their minor child is allowed to have contact with, so if they have to get a restraining order against him to prevent contact, they can do that. The fact that they don't like him is reason enough.
Not unless you've done something criminal. If you are guilty of statutory rape they can report you but if they just want to get rid of you they can get a restraining order.
You would have to prove that the boy was a danger or threat. It is unlikely that his age alone would be sufficient, and if you were successful in getting a restraining order, your daughter could also be legally punished if she violated it. A court isn't a replacement for parenting.
your parents have the restraining order against him so you can still live with him as long as he stays away from your parents.
Of course they can with sufficient grounds.