Thank you for your questions.
1- Both names can remain on title and you can still refinance.
2- He owes you that money anyway so that is not a reason to give up any of your rights.
3- If such an agreement is not part of a Court order, it will be hard to enforce and you would be at his mercy in terms of his claims for repairs, maintenance and improvement expenses.
I do not understand why you pay him rent when you are co-owners.
I think you are correct as to what you speculate his plans are.
You can go back to Court to modify the divorce decree as it relates to the house, and to enforce his obligation to pay over the 401k.
You can get a free consultation from some of the attorneys listed by location here.
Please follow up on this with a local attorney.
I hope this information is helpful.