Is it permissible for my father to pay the expenses of Hajj on my behalf and on behalf of my wife, or to gift us two tickets and the expenses of Hajj, and will our Hajj be considered a supererogatory (nafl) Hajj in either case?

It is permissible to donate the expenses of Hajj to others, and it is not a condition that the money for Hajj comes from the pilgrim himself. If the donation is for a relative, the reward is greater. There is no difference between the methods of donation. Hajj whose expenses are paid by a father's donation becomes obligatory if the person has not performed Hajj before. The donor must have performed Hajj himself first before assisting others. It is permissible to perform Hajj with another's money, such as giving Zakat (definition) to the poor for them to perform Hajj, as this falls under the category of "fi sabilillah" (in the way of Allah) among the Zakat expenditures.