"Are you not having fun?"
Anybody who says they weren't thinking this is a liar.
AM I NOT MERCIFUL?!
Except the extended predicate (~んです) is another whole grammatical beast that this short series couldn't yet hope to approach thoroughly enough for users to gain a meaningful understanding of. Best keep it simple for now, I guess is what they're thinking..
Isn't it essentially just used to emphasize feeling something? I'm sure it has more nuanced meanings but for all intents and purposes wouldn't introducing it like that be fine?
Yes. :D It's more like is this not fun to you + alpha. (Ex. Why, because I think this is fun, or what would you like to do then, we should leave if you're not enjoying yourself, etc. It includes an ulterior motive for the question but the reason is not yet spoken out.)
Isn't「楽しくないですか？」just "Is it not fun?", and wouldn't "Are you not having fun?" be something more like「楽しさをしていないんですか？」(Can する be used that way?)?
"Is it not fun?" should be an acceptable answer as well, since that's more to the letter and less contextual.
楽しさをしていないんですか, however, is not correct. 楽しさ is a kind of 'qualitative' noun (not sure if there's a proper word for that) like "the fun-ness / enjoyability" and is not something that you can する. If you prefer "fun" as a more 'solid' noun, you could use 楽しみ（"pleasure / enjoyment"), which can be used with する, only with a different particle: 楽しみにしています。(usually preceded by a direct object)
Still, LinguDemo's comment (above yours) of 楽しんでいませんか would probably be the better choice.
I checked on Google Translate and both 「楽しくない」 and 「楽しく無い」 seem to be valid writings. Could it be that the first one is more common?
Google Translate isn't exactly an authority on language. I'm not sure if it's technically okay to use the kanji, but in practice it's almost never used.