1. What is this federal minimum wage you speak of?
There are only five states without a minimum wage law of their own (Alabama, Louisiana, Mississippi, Tennessee, and South Carolina).
In addition, four states have laws that set their minimum wage below the federal line (Kansas, Georgia, Wyoming, and New Mexico). However, these states obey federal law and the de facto minimum wage is the federal limit. These states don't need to change their laws because the supremacy clause of the US constitution forced their hand.
If the US Department of Labor were to vanish tomorrow, only those first 5 Southern states would have no minimum wage. The rest would continue to obey the laws that are sitting on the books.
Moreover, you can bet that voters in Kansa, Georgia, Wyoming, New Mexico, AND those first 5 would quickly demand minimum wages closer to the former federal law.
The Federal Minimum Wage is a formality. Real labor laws are made at the state level.
Your state of Arizona has a minimum wage of $6.75/hour, well above the federal limit of $5.85/hour.
My state of Massachusetts, with an overwhelmingly liberal state government, has a minimum wage of $8/hour.
The elimination of the Federal Minimum Wage would effect virtually no one.
Source: Department of Labor