Is it obligatory for one who performs Hajj with his disabled mother to appoint someone to push her wheelchair during the circumambulation (Tawaf) and the ritual walk (Sa'i) for his Hajj to be valid, and does this ruling apply to the rest of the pillars and obligations of Hajj?

Your circumambulation (Tawaf (definition)) and striving (Sa'y) are valid while you push the wheelchair your mother is sitting in. Similarly, your mother's Tawaf and Sa'y are valid if each of you makes the intention. This also applies to the standing at Arafat. As for the stoning (Ramī), deputization is not permissible except for someone who is incapacitated. So, if your mother is unable to perform it, you may stone on her behalf.