I agree that the unseen servant's creator likely must issue the servant verbal commands.
Questions and Answers
Can I order a servant to follow the commands of another party member?
Probably. Although it's a bit of a stretch, the Monster Manual entry on Golem says
The golem’s creator can order the golem to obey the commands of another person…, but the golem’s creator can always resume control over his creation by commanding the golem to obey him alone. (134)
And while similar information is absent from the description of the unseen servant (as well as from animated dead and the figurines of wondrous power and the homunculus and the shield guardian and…), this is convenient enough to generalize into a broader rule about what occurs when an owner orders its creation or possession to obey another.
Can a creature disguised as the creator of the unseen servant issue the unseen servant commands?
I'd rule no. Like a summoned creatures, an unseen servant typically obeys its actual conjurer's commands, not the commands of someone who seems to be the conjurer. Were appearance sufficient to usurp (or even confuse!) a creature's control, I'd want the DM to've made that very clear at the campaign's beginning. That has far-reaching consequences (q.v. golems).
Can a magic mouth issue an unseen servant commands? Can a programmed image issue an unseen servant commands?
As above, I'd rule not in isolation. That is, the unseen servant first could be ordered by its creator to obey commands issued by a magic mouth or a programmed image then would follow the commands issued by those spells, but the unseen servant only obeys its creator by default.
I've speculated a lot here because usually the unseen servant's a treasure-hauler, trapspringer, or toy instead of an integral part of an adventure that demands details; a reasonable DM may disagree with any or all of these points. However, I recommend that if veering sharply from any of these points that the players be informed early lest they be angry when, for example, the unseen servant obeys another because it failed its Perception skill check or Spot skill check to penetrate a disguise!