Is the son obligated to perform the vowed Hajj, I'tikaf, or prayer on behalf of his father, or is it merely recommended?

It is not obligatory for a son to perform the vowed fast or Hajj on behalf of his father. Rather, it is a recommended act. If the father dies having vowed Hajj, and his guardian did not perform it on his behalf, but he left behind an estate, then it is obligatory to take money from it to perform Hajj on his behalf.

As for a vowed fast on behalf of the deceased, it is established in authentic hadiths, and it is recommended for the guardian to perform it on his behalf.

As for vowed I'tikaf (spiritual retreat) and prayer, the majority of scholars hold that neither of them can be performed on behalf of the deceased.