I've never seen or heard of one with a tail.
Skinny pigs are born with a little bit of hair, and lose more of it as they grow. They generally still have hair on their noses, though. Baldwin pigs are pretty much totally hairless.
Skinny pigs were originally a genetic accident. Then the cosmetic and medical research companies started breeding them to produce even more hairless varieties and with poor immune systems so that they were better suited for research. As they became more popular, some breeders did try to improve the immune systems, but there's no way to know whether any given skinny has a compromised immune system or not. While some live to be the normal age of haired guinea pigs, I believe that most die considerably younger.
They're cute as all get-out, but were never intended to be.