Yeah, I sure wouldn't. If a potential tenant filled out with old info and the wrong number, they're gone.
Eh, he's out either way if he can't think of anything else, so there's no risk.
Not necessarily, they may rent to him if he explains the situation.
Well that's why I said "if he can't think of anything else" :) It sounds like he's already offered rent up front and been rejected, so it sounds like time for last-ditch efforts. On the other hand, having already talked to them about that, it might be too late to fly under the radar, they know something's up. But the office may be big enough that the person he talked to isn't the person reviewing his paperwork.
My experience as a renter is that if an individual landlord (like ARS) is not being accommodating, there's a reason, and pushing won't get you anywhere (nor is it worth it anyway because you want both parties to be comfortable). Management companies, however, are often just looking to have their t's crossed and i's dotted. There are plenty of renters, so they can always get someone unremarkable who they don't have to spend any time thinking about. (Come to think of it, they probably have insurance against renters skipping out such that their loss is covered provided they followed industry standard steps in screening tenants.) So if you are a special case, your only defense is to seem like it would cost them more time to refuse you, for example, having a tenant's advocacy group breathing down their neck about whether they're really following the fair housing law. But, that's a last resort, I would try to fly under the radar first.
If the problem really is employment status, you are just going to have the same problem if you try to get a lease somewhere else, so you might as well do all you can to stick with the place where you, presumably, have a good record as a tenant. New York has plenty of independently wealthy people, but they have more coherent tax return history that expresses that, so until then, you'll be limited to places that take guarantors or individual landlords who are willing to work with you.