If Spider-Man meets Batman in an alley and a fight spontaneously erupts he'd make quick work of him.
But Batman's the comic book world's master strategist. He took out the entire JLA by mistake once, based on preparation alone. That's Superman, Wonder Woman, Aquaman, The Flash, the whole gang. If he knew somehow had a heads up that Spider-Man was looking for him, he'd have something ready to squash Peter with by the time they met up.
Or he'd just offer Parker's perpetually broke self a cool million to walk off and "win" that way.
As for Ingrid vs. Sophia... I think Joan Crawford would beat them both.