I'm American. There's nothing wrong with the first sentence but it's perhaps not the strongest way of phrasing it.
"I don't want to own a hedgehog." Would be better. The second sentence is wrong. "They are very stinky."
Americans wouldn't put it like "Have a hedgehog" they'd put it like "Own a hedgehog."
Also they would probably put it like "I wouldn't like to own a hedgehog as a pet. "
I'm not really certain why, both are technically correct but if you don't want to sound Russian in the united states you'll use own instead of have at least in this context.