My question is if the father has relinqueshed his visitation and has not participated in the aspects regarding legal custody, should he still retain legal custody if he is not going to see the child.
Yes, if that's what the current court order stipulates. Remember, a father or mother does not relinquish their rights to visitation, they simply fail to exercise the rights given them by the courts. There is no penalty for not exercising their rights.
As long as contact has been ongoing in the form of letters, phone calls, support or otherwise, there is no termination of any kind.
I am trying to understand the process and what is reasonable. For example, in this scenario, should any change in parenting time be allowed by mutual agreement, when the father has given them up?
Again, he has given nothing up. So the answer to this question is yes. He has every right to petition the court for a change in the visitation schedule.
How does that work in the future when the father has no communication with the child?
Depends on your state statutes. In some states if no contact whatsoever occurs between a parent and a child for one year, and if there is a step-parent in the picture, the non-communicating parent's rights can be terminated, but only if a third-party is ready and willing to adopt.
If there is no such third-party, then no terminatin can or will occur. But this is not a termination in part. This is total termination of parental rights. Including child support.