The question is a little confusing.

Generally, they won't hire you if you cannot do the international trips. There are probably still a few commuter/prop regionals which stay domestic and would hire you.

A US green card and almost any foriegn passport should be fine, I don't think you need a US passport. Might be a problem if you have an Iranian, N. Korean, PLO etc passport, if that would create problems or delays on every trip.

If you are already employed and get a DUI, and cannot go to Canada, they will just keep you off of those trips. I have never heard of anyone getting fired in that situation, but in theory they *could*. I would not want to get a DUI and have to test on probation, that's for sure.

They would also consider the circumstances... one DUI they can probably forgive, and just keep you out of Canada. But if your right to travel in/out of the US got revoked, I think you would be unemployed. Ie you cannot be an illegal immigrant, or have a lapsed visa, or anything like that.