What is the ruling on combining performing Umrah on one's own behalf and then entering into Ihram for Hajj on behalf of another on the Day of Tarwiyah, and what type of Hajj is this, and does it entail a sacrifice (Hady), and is the doer rewarded if he takes a portion of the travel expenses?

It is permissible to perform 'Umrah on your own behalf and then Hajj on behalf of the person who delegated you. In this case, the dam (expiatory sacrifice) of tamattu' becomes obligatory because you availed yourself of the concession of combining the two journeys. It is not a condition that the two rituals be performed on behalf of the same person. If you obtained permission from the one who delegated you, then the dam of tamattu' is equally shared between both of you. However, if you performed 'Umrah without his permission, then the dam is solely upon you. As for the dam of qiran, it is upon the delegate if he was not given permission, and upon both of them if they both gave permission. If one of them gave permission and the other did not, then half of the dam is upon the one who gave permission and the other half is upon the delegate. You will receive the reward for performing Hajj on behalf of this person, and it is permissible to take all the expenses; leaving some of them would be considered an act of benevolence.