If a parent is verbally abusive, can a 14 year old child choose to live with the other parent if they have joint custody?

Your court order will include those details. If you have joint custody, you will continue to have joint custody. If the child wants that changed, a judge must change the court order, until the child reaches the age of majority (18).

A child cannot choose. Only a court can make that decision. However, state laws vary on how much weight will be put on the child's preference.

If a change is wanted, the non custodial parent has to go to court to petition for a modification of the custody order.

As stated by Scott, no, a child can never just "choose" the other parent will have to go to court and prove that a change is needed. The child at 14 can normally testify as to what they want, but all other factors will be included in the court case.