There is nothing by itself that is wrong about you taking possession of a car while they work through the paperwork. Its risky for them, but it also helps seal the deal as once you have the car it is harder to undo the deal.
As far as lying on the paperwork, that I would not do. Lying in order to obtain financing is, at minimum, fraud. If they solicit you to do it and knowingly submit false paperwork, they are culpable as well.
At this point, I would, if you haven't already, return the car and inform them that the only way you will consummate the deal is if the paperwork is adjusted to accurately reflect your income, down payment and residence status. If at that point you are not financeable, so be it. You will be out of the deal and will sleep better at night.
If the dealership give you a hard time, tell them you will alert the finance companies to this fraud and report them to the state. That should get the problem to go away quickly and quietly.
Let me know if you have more concerns.