Since it requires that noone else be facing the challenged, is he/she unable to use it while mounted? Or are they counted as a single unit?
I dont think I understand the question.
but what I think you are asking is, for example: does the cockatrice cavalier's mount benefit from the order challenge, if the character makes the challenge while not mounted?
I don't think the mount benefits at all from the challenge, and I don't think he has to use that challenge mounted at all.
Unless there is some link I'm not thinking of (like shared spell) I don't think the mount benefits from the challenge at all.
I'm not sure what you are getting at with the facing issue?
Challenge: Whenever an order of the cockatrice cavalier issues a challenge, he receives a +1 morale bonus on all melee damage rolls made against the target of his challenge as long as he is the only creature threatening the target (not counting his mount). This bonus increases by +1 for every four levels the cavalier possesses.
In the APG it says, "...he receives a +1 morale bonus on all melee damage rolls made against the target of his challenge as long as he is the only creature threatening the target." My question was, if the cavalier was on his mount at the time, did the mount count as threatening the target as well. As Grick links, it doesn't according to the SRD. Sorry for my lack of clarity.