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While you may have liability under the family expense act, if insurance or medicare pays for most of the bills and there is no probate estate open because his assets are below the probate limit ($100,000) they probably will not pursue you for the difference. Even if there is no insurance or medicare payments, they may not pursue you if there is no probate estate.
Adverse possession is recognized in Illinois but your question really does not pertain to your situation. It applies in a situation where you physically occupy property owned by someone else and your occupancy is open for the owner to see with no objection for at least 20 years. It usually occurs with lot line disputes or vacant land. It does not really apply in a foreclosure situation for a home. If you want to acquire a home through foreclosure you need to attend a foreclosure sale.
The short general answer is yes. Unless the lender gives you a full release for the deficiency, (the difference between the net proceeds from the sale and what you owe) the lender can pursue you to collect on what you owe. While many have yet to do this yet, it is early in this crisis and the statute of limitations on breach of a written contract is 10 years. The lender can also sell your debt to a debt collection company and that company can pursue you at a later date.
In most instances you will have to appear in court for your appointment. Most judges like to size you up and instruct you on your responsibilities as a guardian.
You definitely need to hire an attorney in the state where your father resides. However, under the recent Uniform Adult Guardianship Protected Persons Jurisdiction Act you may be able to get guardianship in your state if your dad has "significant contacts" with that state. The Act has not been passed by all 50 states and because it is so new there is very little case law interpreting the act. If you have to go into a foreign jurisdiction to get guardianship of your father there I would...