"Der Service" is a modern loanword from English and has kept the English spelling, although the final -e is not pronounced. There are two ways to form the genitive case: you can either treat it like a German word that ends in -s which does not take any additional ending --> des Service, or you can adopt the English possessive form (service's) into German and just drop the apostrophe -> des Services. Both forms are accepted. I personally prefer the former one.

"Service" isn't originally a German word - it's an anglicism. You don't pronounce the "-e" when you say it. That's why the rule doesn't apply. Also, in your example, there is a word missing between "hängt" and "der". ;)