Is it permissible to intend sponsoring an orphan on behalf of one's parents, the sponsor, and his household, and does the reward include all of them, or must the intention be for only one person, and do multiple people who jointly paid the sponsorship amount receive the reward?

It is permissible for you to intend, by your sponsorship, the reward for yourself, and to dedicate some of it to whomever you wish, be they living or deceased, and the reward will encompass all of you. If several people share equally in sponsoring an orphan, then each one receives the same reward as the other, unless one of them has a special role, such as caring for the orphan, and so forth.