She was to refi immediately, he was to sign the quit claim within a reasonable length of time.He stated that she was to refi immediately and he was to give her the quit claim immediately. Therefore it could be "argued" that they were to happen simultaneously. I am not saying that the order specifically says that, I am saying that it could be interpreted/argued that way that therefore I think there is some doubt that a judge would consider that "willful" contempt. Particularly since he will be able to honestly state that he would have cheerfully signed the quit claim deed at the closing table of her refinance.
And, no, she wouldn't be selling him the house for a dollar. Since she was awarded the house in the divorce she would be selling him the house for the balance of the mortgage due, plus even perhaps some equity if any equity exists. These things go through a normal closing just like any other sale. There is nothing informal about it.
If she won't agree then he needs to take it to court, but if she does, its no different than any other voluntary sale of property and doesn't require any court involvement.
The most important thing however is that he cannot just move into the house, and take over the mortgage, without eventually having a closing happen or court intervention happen. Otherwise, he simply becomes her long term tenant and she continues to own the home via the divorce decree. He could end up paying it off in whatever amount of years remains on the mortgage, with her continuing to own it via the divorce decree. That of course would NOT be in his best interest, other than protecting his credit.
His signing the quit claim is contingent upon her refi. That's how it should play out.
However, when dealing with incompetent lawyers and judges, anything can happen.