J.S.
Hello C.. I just wanted to let you know you are not alone. My 3 yo daughter is also a bolter that cant sit still. She will not sit in a stroller but maybe 2 minutes. A shopping cart...forget it. She screams at the top of her lungs if I even try to put her in one. I let her push the cart with me instead, or the stroller, whichever it may be. That doesnt always work though. I also let her walk away a bit. I tell her she needs to stay where I can see her and she can see me. She walks a little ahead of me and feels like she's leading the way (even though I tell her when to turn or stop)
I dont really have any advice on the other two issues you are having, but I wanted you to know you are not alone, you are not a terrible parent that cant control your child. He is at a stage of finding his independance and pushing limits to see what he can get away with.
Dont worry about what other parents think. Sometimes others are looking at your situation thinking "I remember when that happened to me" When my daughter is throwing a fit in the store (like when I try to put her in a cart cuz Im in a hurry) and people start to stare, I just smile at them and keep on with my shopping. You probably dont notice other parents going thru this cuz you are focused on your son and getting him to listen to you. When your son is on really good behavior is when you'll notice other parents trying to find their kid in the clothing racks. At least thats how Ive found it to happen :)
Good luck and remember you're not alone! Dont be so hard on yourself.