If my father's mortgage company wants $40,000 for me to keep his property or they will foreclose but the deed is now in my name, am I responsible for the mortgage?

if you did not sign for the mortgage you do not owe the debt.  the mortgage company can foreclose and sell the house and that is all.   you could do a chapter 13 bankruptcy and cure the mortgage over 60 months if you really want to keep the home.