Public Charge and Financial Sponsor

1. Yes, Negotiate with the department, or let them sue, then negotiate with the lawyer.

2. Yes, she has demonstrated that she is likely to be a "public charge", read this carefully http://www.uscis.gov/green-card/green-card-processes-and-procedures/public-charge

3. Depends on where the money county pay out of, you might or might no be responsible

You signed http://www.uscis.gov/i-864 , did you not? On page 7 of 9 of http://www.uscis.gov/sites/default/files/files/form/i-864.pdf it clearly listed what program you are responsible for reimbursement, and which program you are not responsible.