LLC valid in whole US
A company registered in one U.S. state is recognized in all U.S. states and territories. It still doesn't mean you can conduct business legally in other states without having to register this company as foreign entity (also called "foreign qualification") in those particular states, but this is important only if you create strong nexus in those particular states (example of strong nexus is having an office, etc.). If you want to open a bank account, or sell to clients in other states, you wouldn't need to foreign qualify in these states.
That being said, if you register your company in Arizona, and someone in Ohio would register a company with exactly the same name, there will be little you can do about it (unless you have already established meaningful brand, but then its a question for IP attorney to deal with). To protect your brand name for use in all of U.S. registering your company name as a Trademark is not a bad idea at all. You do it with USPTO and it get's you covered nationwide. We offer this as an additional service in our company registration package.