He throws a tantrum if I don't let him run. Also he likes to touch everything "and take them out of their place. It's ok in the mall where I can put it back, but it frustrates M. sometimes."
So -- let him throw the tantrum.
Time for him to learn. You're out. He runs. You stop him. He throws a tantrum.
Then you tell him firmly, "We don't run in the mall. When you run here we go home NOW." And you pick him up -- do not try to make him walk --and haul him to the car. Yes, haul him while he screams and kicks. Strap him in as you say,"When you run at the mall, we leave NOW." And nothing else! Let him burn out his tantrum. And then do it all over again next time. Tell him BEFORE you get in the door of the mall: "We're at the mall! We'll walk and look in windows. But if you let go of M.'s hand and run, we will leave." Repeat it as you get him out. Repeat it as you start to walk.
If he runs, stop him. Tantrum. Firm but calm removal. Not fun for you, no, but it's past time he learned that this behavior means instant removal. No second chances and certainly no toleration of the running away. Repeat as needed. He must, must get this kind of firmness or he's going to keep running away. Don't scream or shout, just do it, no matter how many people stare at you.
And why is he picking up stuff and moving it around? "Dont' touch" is what I hear constantly in the mall from moms with toddlers. Occupy his hands or, if he touches, into a stroller he goes, instant strapdown, and yes, removal if he has a tantrum.
I'd skip the mall for outdoor play much more for this active guy, but it's winter and I know that's not doable in Chicago. So he needs to learn some mall manners starting now.
Also, you tried the leash once for 15 minutes and gave up. Too soon. Try again. And you're not willing to use the stroller because he fusses in it, I'd bet. You're going to have to endure his displeasure for quite a while in order for him to learn anything; you're going to have to endure the looks from strangers or the tut-tutting of others if you want him to gain some self-discipline while out with you. Fussing, touching, running -- into the stroller or right out the door and home. And I'd try the leash again. Make it a game if you possibly can and use it with him at home or around the block and see if he'll be your "pony" or "puppy" so it's fun, before you take him to the mall with it.
But whatever you do, he doesn't need cartoons on getting lost because he's still too young to understand that idea -- he will tell you "Yes, M., I don't want to be lost" but he's only pleasing the adult; he doesn't really have any concept of it because he's never actually been lost except when he wanted to hide and knew you'd find him. Give up on teaching him about "lost" and focus more on instant and firm consequences -- mostly, immediate departure -- when he does these things.