Why my dictionary shows for "bye" "tschüs" - only with one "S"? So what is correct?
Both spellings are correct.
Use tschüs if you pronounce the word with a long vowel; use tschüss if you pronounce the word with a short vowel.
Well it is a friendly way to say goodbye that in past was used only between young people but today is very common in people of all ages. I found this answer on youtube: the channel is "tedescopertutti" see lection "come salutare" (2 or 3 I don't remember); unluckily for you this is a course in Italian but you may have a look. For all I found interesting this: almanii.blogspot.com there you can find a lot of free resources to learn German or to improve it. Sorry for my not perfect English (I'm Italian and I study myself). :-)
It's from a Romance language - related to Spanish adiós.
An earlier form was atschüs.
It's a [y] sound (IPA),. It's not a "oo" or a "ew". If you can pronounce the French "u", it's the same.
There's also a soft "t" sound at the beginning. "T'sheuss". The "eu" is not the ü, but it's closer.
Good bye is a naff old fashioned phrase for parting company. "Cheers" is the present day way of saying goodbye - rather dismissive perhaps but at least as popular as "bye"