Is it permissible for the father to back out of his verbal promise to sell the house, after having received part of the payment, due to rising prices and the delay in legal proceedings, knowing that the buyer (the aunt's husband) had previously withdrawn from the purchase then returned and affirmed his desire?

What transpired between your father and your aunt's husband was not a promise of sale, but rather a voidable sales contract due to gharar (definition) (uncertainty) and your father's inability to deliver the sold item at the time of the contract. Therefore, your father is permitted to withdraw from it. If the currency's value has significantly depreciated from the time your father received the money until his withdrawal, then they must share the damage by compensating for the decrease in value.