Are “Kennedy Steve’s” various jokes allowed in ATC communications?
(US-based answer)

Yes, it's legal. Yes, everyone - including pilots - is held to the same standard of professionalism. And there are no real consequences for anyone (usually). The simple fact is that we're all human beings and humor, slang and jokes are all part of how we communicate and even handle stress.

Legally, the FAA doesn't regulate radio communication, the FCC does. Of course, the FAA has aviation regulations and guidelines that apply, but they're very general (as they should be). Using standard phraseology is strongly encouraged at all times for clearer, quicker communication. The ATC orders 2-4-14 say:


  Use the words or phrases in radiotelephone and interphone
  communication as contained in the P/CG or, within areas where
  Controller Pilot Data Link Communications (CPDLC) is in use, the
  phraseology contained in the applicable CPDLC message set.


In other words, stick to standard phrasing. But there's nothing in there about avoiding humor or jokes.

As for different controllers, they all follow the same standards for radio use no matter where or on which frequency they're working. In my experience, the quieter the frequency the more likely it is that someone - controller or pilot - will joke or even have a brief conversation. I was once taxiing a C172 to the far end of a 12,000ft runway - which takes some time - and the tower controller asked me a few short questions about my accent, where I was from etc. But I was the only aircraft on his frequency at the time and I'm sure he wouldn't have done it if it would have impacted operations or safety in any way.

Which brings me to your final point. In normal circumstances, no one cares about this stuff at all because a) it has no significant impact on anything, and b) it makes life a little more interesting for everyone. It's practically impossible to fly cross-country for more than half an hour without hearing someone make a joke, chat with someone else (briefly), or even complain about something. And again, in normal circumstances that's completely fine: no one expects pilots and ATC to act like robots with a limited vocabulary.

But, if a joke or non-standard phraseology causes offence or a misunderstanding or - in the worst case - an incident of some kind then the FAA or NTSB could be interested. This column from NASA's ASRS team has some examples of reports of problems causes by non-standard phraseology, including this one:


  "I called for clearance to St. Louis as follows: 'Clearance delivery,
  company ident, ATIS info, federal aid to St. Louis.' Federal aid was
  meant to mean FAA clearance in a joking fashion. The Controller
  misinterpreted this to mean that we were being hijacked and called the
  FBI and airport police...I used no 'standard' phraseology to indicate
  nor was it my intent to indicate we had a hijacking...I will use
  absolutely standard phraseology in the future..." (# 248982)


And this incident got quite a lot of attention, even outside the aviation press. I don't think it makes sense to say that it's 'more serious' for ATC to make jokes than pilots because every incident is different anyway.

All in all, joking on the radio is like joking in any professional context: if it doesn't disrupt your work or other people's work, if you're not insulting people or wasting their time, and if you're not saying anything stupid or dangerous then no one really minds.