Debatable. Shaggy Man might not take any of them out, but since the JLA couldn't hurt him enough to kill him (he basically has herald-like durability and heals instantly from anything) it's doubtful that this crew could either. Shaggy Man takes out the weaker Avengers. Then it's just a stalemate.
The only possibility is the Avengers finding some way to contain him (Quasar?) and sending him into space, effectively a win via ring-out. I could see that happening eventually.
No that's Doomsday. To my knowledge, SM can't evolve or anything.
I liken Shaggy Man to Juggernaut. He's got insane durability and healing, so he's basically invulnerable. The JLA (Superman, MM, WW, etc.) can't take him out. But strength-wise he's not beyond your average herald-type, and his speed is probably around Hulk-level (give or take).
His last appearence the JLA teleported him to some asteroid belt, out of reach of Earth. He didn't die in space either...he was just stuck.
C'mon people what about the Scarlet Witch she can whip up a hex that might be able to render his invulnerable body vulnerable, if this happens it's a wrap, Sentry, Thor, Wonderman,Iron Man will then bounce him out of there like a superball, and if not Thor can open up a pocket dimension suitable just for him with Scarlet Witches help to make his stay more pleasurable (imagine if they managed to send him to the place that Asteroth came from? you know that nasty chick from the Stormbreaker mini? Whooa he won't be invulnerable anymore I bet!), when I think of it, Shaggy Man would be a minor nuisance to these particular Avengers.