Handicapped Children and Others

I have read of cases where a parent is charged with killing his (or her) severely handicapped child. I think the parent's motivation is exhaustion from constantly caring for the child with hope of an end to the task.

I have also heard of parents with severely handicapped children who give up the child to an orphanage or other institution.

My question is, do parents have a legal right to give up their children?

I've read of numerous cases where divorcing parents fight over "child custody." In every case I have read about, each parent wants custody of the child(ren) to the exclusion of the other. Are then any cases where each parent tries to persuade the court to assign the child(ren) (handicapped or otherwise) to the other parent?