The mother can still have sole legal and physical custody when the father is awarded visitations. Custody and visitations are separate matters. The mother would be required to obey the visitation schedule.
by whom?
They can be held in contempt which is grounds for a change of custody.
The non-custodial parent should file for visitation rights for the child in the county where custody was given. If the parent who has custody of the child is preventing the non-custodial parent the right of visitation they can be found in contempt of court if visitation has already been established. If the custodial parent has too many repeated contempts filed against them, custody can be switched to the other parent.
Generally, the mother gets custody in Tennessee. Unless there are some horrible, provable reasons that the mother is unfit to have custody. I have been through this very situation myself. Tennessee is a Mother's Rights state.
He/she can file a motion for contempt of court, and if granted, a change of custody. I teach parents how to collect evidence and how to do this without the need of hiring an attorney. see link
Leaves them permanently or in violation of court ordered custody/visitation time? In either circumstances, the non-custodial parent may file an action for contempt of court against the custodial parent and/or file for custody/visitation modification based on the same.
If you refuse court ordered visitation then you would be considered in contempt of court and a judge may issue a warrant for your arrest to face the charge. It could cost you money, or some time in jail and money.
The child can suggest perhaps, but the final decision rests with the couple and the judge. What usually happens is that couples will have Joint Legal Custody, but one parent or the other will have Primary Custody, leaving the other with Visitation Rights.
If you live in the US... Yes, Dad still has custody, unless he has voluntarily given legal custody to someone else or the court has awarded legal custody to someone else (normally only happens in cases where the parent has been proven to be unfit).
He/she can lose custody all together. The non-custodial parent needs to file a motion for contempt of a court order. If the custodial parent continues to violate the order they could eventually lose custody.
If the parents are unmarried and there is no custody order then nothing happens. The father needs to establish his paternity legally and request custody and/or visitations. The court will also issue a child support order. If there is no marriage to establish legal status then the parties must establish legal relationships and custody by court orders.
The child is always free to express their wishes, either in court or by writing a letter to the judge in charge of their custody/visitation. If the judge feels the child provides valid reasons why they no longer wish to visit their father, the visitation order may be modified. However, court ordered visitation must continue until that happens, if it happens. Otherwise, the mother could be found in contempt of court and fined or even jailed for their child's failure to obey court ordered visitation.