Is it permissible to swear to pay someone seven thousand dollars in exchange for expediting naturalization procedures, and is this oath religiously void, knowing that the شيخ (sheikh) first issued a fatwa that the briber and the bribed are in Hellfire and the contract is void, then recanted and issued a fatwa that expediting procedures is permissible, and compelled the father to pay?

You made a mistake by agreeing to expedite procedures in exchange for money, as it might be a forbidden bribe. "The Messenger of Allah (peace be upon him) cursed the one who gives a bribe and the one who takes it." However, if your file was unjustly removed and cannot be returned except by paying a bribe, then it is permissible for you to pay, and the money is unlawful for the one who takes it. As for your father's oath to pay the amount, it is a binding oath, but he is not obliged to fulfill it. Rather, he should break it and offer expiation for his oath, based on the Prophet's (peace be upon him) saying: "Whoever takes an oath and then sees something better than it, let him do that which is better and offer expiation for his oath."