Good questions
* This is in no way legal advise, just a little informative post. I am not a lawyer or legal spokes person. The question was pertaining to state law not local ordinances.
Criminal Laws: Title 22 Chapter 14
Traffic Laws: 32-20-6.6
1. Trunk guns are good to go (or at least not regulated). This obviously pertains to citizens legally able to possess said firearms. I am assuming you definition of loaded firearm (long gun) is loaded mag, chamber empty. I caution people who have a round in the chamber for long guns when it is not in your physical control.
2. Mule / ATV - typically firearms must be cased, and unloaded unless specified in some exceptions in above statutes. Visible gun rack, not regulated.
Edit: There are more administrative rules from Game, Fish and Parks pertaining weapons. Adminstrative rule carry the same weight as law. See their website under regulations for more information.