Do I have legal argument for the unexpected change in title and duties or ability to break the payback agreement as a result?

No one would have any idea without reviewing the contact to see all the aspects of the agreement.

HOW CAN I SIGN THE LEASE FOR HER AND MYSELF

thats tricky you can contact the facility and ask them to help you but if she is incompetent then you cannot so you may have to file for...

it depends on a state police check yes if will come up. you should talk to a lawyer

It sounds like you are judgment proof already

You can't really say if this ok or not without reviewing the whole document.

Can I be legally responsible for the $11,000.

I am not sure what is going on here. You are being sued for defaulting it sounds like so you should talk to a lawyer.

Can I take him to small claims court and try to recoup some of my money.

I suspect if will find itself in small claims anyway he will likely sue you as well. Then you can file a counter claim describing his bad work.

How much would it be to take on a case like this? Will he be able to get some type of free consultation prior to hiring a lawyer

A lawyer here on this site cannot recommend you hire them. However you can always get a free consultation from a lawyer on this site. He is facing...

A lawyer on this site cannot recommend someone however take advantage of AVVO and speak to a lawyer for a consult

What can I do if my 7 year old doesn't want to stay with her dad every other weekend?

Yes only court intervention would help her in this matter.