As the saying goes, you marry one person and divorce another...(o; - here are the responses to your questions....
"My husband's ex-wife continues to violate their divorce decree/order by not allowing him access to his son on his scheduled residential times"
He can file a contempt motion with the court. In many/most states, the police are not going to be involved in enforcing a civil order, like visitation. He chose this woman, it's his problem to enforce his rights. --- if he files the motion for contempt and gets an enforcable order signed by a judge, next time she refuses access to the child, the police say then they can assist.
"bad mouthing him in front of the child"
Hard to enforce without dragging the child into it.
-- the divorce decree states that either parent will refrain from making disparaging or negative remarks about the other parent in the presence of the child. What is the point of that being in the document if it is not enforcable?
"having adult conversations in front of the child"
The government really has no business telling a parent they can't do this. ---- the divorce decree states this is not allowed
Custody orders often contain 'feel good' orders. The court WANTS people to follow the order because it is there, but doesn't expect to have any real ability to enforce the order.--- understood, but confusing to the parents as to why this would be in a divorce decree if it is not expected to be upheld
"His ex has kept his son from him on scheduled times on three separate occasions now."
Only three times? How long have they been apart? How long have orders been in place? ---- 24 months and 24 months
"Is there anything my husband can do about her false claims of abuse?"
Normally, no. The government doesn't want anyone to be discouraged from reporting possible abuse. --- that is disappointing, any fool can call and make a false claim and waste the taxpayer's money and the investigator's time when there are real abusers out there....
"There are no rules or structure at the her home and at the his home, there are rules and discipline."
Why did he have a child with a crazy lady?---- well, she wasn't crazy at the time of conception, you marry one person and divorce another, accidents happen, condoms break....and you end up marrying someone to try to do the right thing.....
"Can an attorney send a cease and desist letter to the ex stating she is in contempt and will be taken to court if she does not stop or should we just take it to that next step and get ready to go to court?"
A cease and desist letter doesn't change anything.
"How difficult is it to prove contempt in court?"
Depends on what is being alleged.----what is being alleged is she is breaking the divorce decree by not allowing access to the child on the father's residential times - three times, as well as breaking other rules within the decree (bad mouthing in front of the child, etc., as noted in the original post)
"Do we have enough to go to court that a judge would consider all of this to be contempt?"
Most of this will depend on what the kid says, likely.---- why does this have anything to do with what the child would say? I don't understand why the word of a 12 year old would take precendence? Please explain
"What punishment would the ex receive for being in contempt? "
Probably a tongue-lashing, the first time. Contempt can result in jail, potentially.