Is it permissible to appoint someone to perform Hajj on behalf of a deceased person who intended to perform Hajj before their death? Is it a condition that the proxy be a relative? And should the sacrifice be in the name of the deceased or the proxy?

If a Muslim dies after being able to perform Hajj, then their heir must perform Hajj on their behalf themselves, or appoint someone to perform Hajj on their behalf from the deceased's wealth; because it is a debt upon them, and it is most deserving of being fulfilled. It is not necessary for the appointed person to be a relative. The sacrificial offering (Hady) will be in the name of the deceased, if it is a requirement of Hajj. However, if the Hajj is ifrad (single Hajj), then no Hady is obligatory unless the appointed person commits a prohibited act therein that incurs a redemption (fidya).