Most expats I know of maintain a residence address in the states, my father's been working in iraq and kuwait since 2003 but still is considered an AZ resident because of owning a house here, keeps his commercial drivers license up to date here, (and pays taxes here). If you don't have a house then use a relative's address as your 'residence' while you are working overseas, just get them to forward the important mail to you. If you're working for the military you can also get mail through the APO which should be no problem for getting non-resident permits through. And like you mention, the only requirement for AZ CWP is US Citizenship or permanent residency and it's a shall issue state, I don't see a foreign mailing address as being a problem.
As far as the tax exemption for remaining outside the states it doesn't void your residency, you still have to file your tax return and the exemption only covers up to a certain amount (around 80k I don't recall the exact amount) after which you still have to pay your federal and state taxes, and you've got to pay your Social Security/FICA in any case. You can't be a resident of NO state, I believe even if you have no residence in the US it defaults to whichever state you last resided in, the tax man always cometh.