No, an Iowa marriage license is only valid in Iowa. You need a license valid in the location where the marriage is to take place.
There is no requirement to do so. Your marriage will be recognized.
No, you need the license from the location of the ceremony. You will have to apply in New Mexico.
No, it would not be legal. The license must be issued by the country, Mexico, where the ceremony is to take place.
You have to get married in the same state in which you obtained your marriage license. But you can apply for an Arizona marriage license if that is the state in which you have decided to get married. Just because you got a marriage license in one state doesn't mean you can't change your mind and apply to get married in a different state. But when you do get married, you have to have a marriage license in the state in which you get married. The marriage license is not recorded as a "done deal" until after the wedding, so it shouldn't be a problem.
Yes. A certificate of marriage, regardless of where it was issued, is sufficient proof of legal change of name for a New Mexico driver's license. The gender of your spouse is irrelevant.
You need a copy of your marriage certificate from Mexico. That is your proof that you are married.
Contact the place that issued the license. That should be the court house. And what you probably want is the Certificate of Marriage, not the license.
Maybe, if you believe hard enough
That would be bigamy which is a crime. And to get the license, they would have to lie on the application, which is also a crime.
It is written into the US Constitution. States have to recognize the actions of other state.
If your marriage license was lost in the mail, it is not a big problem. You can always get a copy at the courthouse where you applied for the license.