Cats survive, so would little dogs. Some may be overbred but rat terriers would do fine anywhere rodents live. Little animals find little hiding places. Some of them do have a tendency to make too much noise, but it won’t take many generations to weed that gene out.
The largest dogs may problems catching enough food, they may not be fast enough to bring down large game. Size-wise medium size dogs in the 30-50 pound range probably have the advantage for lone animals. In packs larger dogs could go after big game. Some larger dogs may end up breeding with coyotes and wolves and keep their genes around.
Altogether, the environment will make the difference. Dogs are smart and adaptable, the right climate and available game should give almost any dog a chance.
The poodle-type coat doesn’t shed. I have a poodle mix and his hair is a disaster. He’s constantly got burs and twigs stuck in him. He once got a bumblebee stuck in his fur. Lucky the bee was hung upside down so the stinger was facing out. Freaked both of them out but I was able grasp the bee on the sides and work him lose, after which he Noped off away from us.
I get my dog shaved down as often as possible. He needs a haircut now, actually, but he was just in the hospital and I haven’t wanted to send him. He needs his vax updates first, too.
As far as wild dogs - they tend to revert back to a sort of dingo looking dog if they’re allowed to run and breed indiscriminately. Presumably, that’s the most successful form for a dog.
English bulldogs, as they are now, would not survive in the wild. They have difficulty breeding without human intervention, have such short muzzles that they overheat rapidly, and have extreme skin/allergy issues. They’re light years away from their origins.