I work at a wildlife rehabilitation clinic and I've seen first hand opossums hanging upside down by their tails. They can do this when young, though are too heavy as they grow.
It's a common misconception that opossums hang upside down-- they do not. Sloths commonly hang upside down, and many monkey species are known to hang upside down, usually as part of play, but not for long periods of rest like the bat. Bats are the only flying mammal to hang upside down when they rest.
No- there are sloths and those possum things