According to the precedents set by Anderson v Anderson, there are a list of factors when deciding custodial arrangements of children. Chief among them is the qualifications and fitness of the parent and their ability to control and direct the children. If the underage mother is able to provide this, then she will retain custody.
This is an interpretive area, usually without a clear direction of the law. The legislatures have not sufficiently addressed the issues of underage pregnancy, other than to abolish the requirement of being sent to a home for unwed mothers. Which still exist.
sole control and custodysee link
Unless visitation rights for the non-custodial parent were allowed in the divorce paperwork, the custodial parent is completely within their rights to deny the non-custodial parent visitation....however, the non-custodial parent may sue for visitation rights.
If the court has awarded you visitation rights, then you have those rights legally and they cannot be denied by the custodial parent.
see related question
Police don't question mothers about custody when there's a claim the father kidnapped the child, but single fathers have absolutely no rights to the child until granted them. see link
Parental rights are paramount to "grandparents rights". In most jurisdictions there are no such rights.
It depends on what kind of custody arrangements you have. If you have joint legal custody, you both have to decide together what is best for the children. If you do not want the child to have a tattoo, the non-custodial parent should not allow the child to have it done and you can file contempt charges if they do.
no, that's custodial interference
What rights do you want?
YES
They need to file for custody