Can my father pay for me and my wife's Hajj? Or can my father gift me the airfare and expenses for our Hajj? If either way will our hajj be nafil.
There is no reason why a well-wisher should not donate Hajj expenses to someone, and it is not a condition of obligatory or naafil Hajj being valid that it should be paid for with the pilgrim’s own money.
If this donation is given to a relative, then the reward is greater, and he will have a reward like that of Hajj, in sha Allaah. What your father is going to do, paying the expenses for Hajj for you and your wife, is something that is to be appreciated, and he is included in the words of the verse (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety)”
There is no difference between the two ways mentioned in the question. If you have not done Hajj before then this Hajj will be the obligatory one, and it will not be naafil just because your father donated the expenses.
In the answer to question no. 36990, we have quoted a fatwa from the scholars of the Standing Committee concerning this. Please refer to it.
Your father who is going to donate this money should have performed Hajj already, because the command is addressed to him too.
It is not right to delay Hajj then appointed help someone else to do Hajj, rather he should hasten to do Hajj himself, then if he has any extra money he may help you to do Hajj.
In the answer to question no. 36637 we have quoted a fatwa from the scholars of the Standing Committee concerning this. Please refer to it.
One indication that it is permissible to do Hajj with money other than one’s own is the fact that it is permissible to give zakaah to a poor person so that he can do Hajj. This is included in the verse in which Allaah says, listing those who are entitled to zakaah, (interpretation of the meaning):
“and for Allaah’s Cause”
that includes both jihad and Hajj.
This has already been explained in the answer to question no. 40023.
And Allaah knows best.