Is the questioner obligated to pay rent to his wife's father, who previously stated that the apartment was for his daughter and he would not pay anything, then refused to take rent, and is now bargaining with him for his possessions in the house in exchange for rent?

Since you offered the rent and the owner refused, this means he lent you the apartment. Therefore, he has no right to take anything from your belongings in exchange for your dwelling in it, because the lender is not entitled to rent for a loan. If he wishes to terminate the loan, he must inform you. Disputes are to be referred to the judiciary.