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You'll certainly want to ask them this. Steer clear of any company that is not insured in Arizona, because that means that any incidental damage they may do on your property will be a real hassle to recover compensation for. It is never wise to work with uninsured tradesmen of any kind. Most states also require that someone have insurance in order to be issued a trade license. If they say they're licensed but not insured, they're being less than truthful.

This is very important, since an estimate is just that. Many contractors will submit a "paper estimate", meaning they've written the estimate down on paper. The bill is the true amount owed and will almost always say "Invoice" or "Bill" at the top of the page. The bill should also list the date it was issued, the due date of payment, and a list of services rendered.

This really only matters if you have a preference, but it is good to ask. If you have a strong ideological belief about unions or nonunions, it's perfectly acceptable to make a decision based on the preference.

Some companies may require a 10% deposit, some may not require any. For larger projects, you'll almost always need to give a deposit, and it's good to ask how much the deposit will be before going forward with a project. The Phoenix contractor will know how much they need for a deposit when they've given you the estimate, so ask early.