This is all my very non lawyer understanding.
1. Public schools in CO are gun free zones, you can keep your handgun out of view in a locked vehicle in a school parking lot so I assume car carry is okay on the road right next to it. Private schools, I think, can make any rules they desire so you'd have to know but I treat them like public schools (although the possibility could be that they are even stricter) but you don't plan on being on the "real" property of said schools so it shouldn't be a problem.
2.Your car is your real private property and is like a rolling home. Police need a warrant to search it, and open and conceal carry and acceptable without a CHP (except OC in Denver). Driving by a school would be of no consequence.
3.Most Stores own the parking lot in front of them, so they can do whatever rules they choose.
4.You're fine, I think.
You can conceal carry your handgun in your car everywhere in Colorado, you can open carry in your car everywhere but Denver. I would say if you're under 21 just keep your gun locked in your glove box if it's not physically on your person. Police need a warrant to search it regardless. Just remember to not keep your license/registration ect. next to the gun, police won't like that if you get pulled over and you have to touch/move a gun while they are around.