Original comics: Jughead's family not rich. Don't remember them saying how he got the nickname though.
It was literally because his head was shaped like a jug -- tall and cylindrical with ears that stuck out like handles. In the '40s when the character was created, it was also a slang term for a donkey or a foolish person. So it's a nickname akin to "Goofy" or "Dizzy" or something like that.
Cheryl Blossom's parents are normal, even nice. Both kids though are utter spoiled brats.
Also, Jason isn't dead.