You'll find Colorado a great place to open carry. If you like, we can meet up when you arrive in Colorado, and I can show you around with a little OC thrown in to help you get going here. I was in Oklahoma (Sapulpa) back in May, and I even open carried while there. Yeah, the OC bill had passed and been signed by your governor, but it still wasn't really okay until this coming November, but I did it anyhow.
The answers to your questions:
1. If you own a handgun, and are over 18, you can openly carry your sidearm here. A concealed permit is not necessary for OC.
2. A private vehicle is considered as an extension of your home. You can carry both openly or concealed in a private automobile.
3. Yes it is legal to open carry in the national forest. You can't carry in buildings, such as a ranger station or visitor center. They should be marked to that effect.
4. Ordinarily you can shoot in the national forest. They have areas for that. HOWEVER!!!!!!!!!!!!!!!!! Due to excessively dry conditions, shooting has been disallowed until we get sufficient moisture to make a fire unlikely. Check with the ranger district offices, but last I heard (Saturday) was that shooting firearms is currently prohibited.
5. It would be better, and more secure, to leave it in a glove compartment or locked in your trunk. That being said, you can leave it under a seat, but I don't feel that's very secure. Just my personal attitude toward firearm security. Very few locations will require you to leave your sidearm in the car. Some malls are posted 'no firearms'. Others are not. I don't tend to frequent malls anyhow, so for me it is somewhat a moot point. I carry everywhere that isn't posted. There are very few locations that are posted. Some are obvious, like school zones, post offices, or any location with metal detectors and security types in place.
Hopefully this will answer your questions.
Welcome to the Colorado Front Range!