1. Yes. If they ask you to leave, and you have a manager tell you it is company policy to prohibit weapons, then you must leave. If you fail to do so you can be charged with trespassing.
2. No. The state preemption law regarding storage only covers public (government-run) facilities and events, and even this is not widely known nor enforced - some facilities do not have lockers at all.
3. It's not a question of superiority, but one of preference. Open carry is easier in AZ because of the weather, it has educational benefits for the public (getting them used to seeing their peers carry weapons peacefully) and it has a deterrent effect on the criminal element (bad guys usually avoid attacking armed targets.)
- Self defense.
- Education (see #3)
- Deterrence (see #3)
- Civic responsibility.