The suit is in the comics, starting with Amazing Fantasy #15. And what "realistic thing" are you talking about? There's nothing realistic about Spider-Man's world. There is no suit which would be "realistic" for him. He's an agility-based character and he wears a costume that makes it easy for him to move, similar to any acrobat. Because this is fantasy he's given a colorful costume with a spidery theme to it.
He doesn't need running shoes because:
1. Heavy running isn't apart of his normal activity (Unlike, say, the Flash). The kind of soles on the ASM suit are if you're running or jogging for miles at a time.
2. It would logically interfere with his natural climbing ability, which while never having been thoroughly explained (Although to correct someone, the bards in the Raimi films have been mentioned in the comics), regardless is generated from his skin. And there's no way he'd be making any useful contact through those thick soles.
3. In the comics, the boots have always been portrayed as having a thin layer at the sole, not as being merely stockings.
4. Being superhuman Spidey's muscle and bone structure would have to be more dense than a normal human. He doesn't need the same kind of protection.
