Is it permissible for me to perform Hajj on behalf of my grandmother who is unable to do so, knowing that I have already performed Hajj for myself, and she has performed Hajj for herself, or is the proxy (niyabah) only for someone who has not performed Hajj? And is it permissible for me to dedicate the reward of Hajj to her? And is it permissible for me to perform Umrah and then dedicate the reward of the Umrah to a living person who is able to perform Umrah?

Your Hajj on behalf of your grandmother, who is unable to perform Hajj herself and has already performed the obligatory Hajj (Hajj al-Islam), is permissible. As for Umrah on behalf of your father, if he has not performed Umrah himself while being able, then it is not permissible for you to perform Umrah on his behalf. If he has performed Umrah himself and is capable of performing it personally, then there are two opinions regarding this, and the more cautious approach is to refrain from doing so to avoid disagreement. It is not permissible, by consensus, for someone who is able to perform the obligatory Hajj himself to appoint a proxy for it. Voluntary Hajj is divided into three categories: First: Someone who has not performed Hajj al-Islam cannot appoint a proxy for voluntary Hajj. Second: Someone who has performed Hajj al-Islam but is unable to perform Hajj himself, it is permissible for him to appoint a proxy for voluntary Hajj. Third: Someone who has performed Hajj al-Islam and is capable of performing Hajj himself, there are two narrations concerning this: one permits it, and the other does not. If you perform Umrah and then dedicate the reward of the Umrah to your living father, this is permissible according to the Hanafi and Hanbali schools of thought.