The difference between Present Perfect Simple and Present Perfect Continuous in grammar books stresses whether the action is either a) uninterrupted or b) unfinished.
1) I have written three letters.
2) I have been writing three letters.
What is implied is that in sentence 1 three letters have been completed, whereas in the second sentence though writing has started on 3 letters, none has been completed yet (and this activity may still be in progress).
In the negative, the same would apply. Sentence one would mean that 3 letters haven't been completed, whereas sentence 2 would mean that the act of writing 3 letters (none of which has been completed) has not taken place.
Another aspect the Present Perfect Continuous focuses on is that it is used for repeated activities. This is where your example about work might apply. If the statement says "I haven't been working since my baby was born", it simply stresses an activity which is repetitive (i.e. you go to work on a daily basis) versus the statement "I haven't worked since my baby was born" which states a fact and doesn't stress the repetitive nature of work.
As for your question about which is more common for native speakers, we'd have to do a frequency check and still that wouldn't prove anything since speakers don't necessarily use grammatically correct English when they speak (how many times do you hear "who" when it is the object of a relative clause instead of "whom"?), which is acceptable in certain cases.
Finally to answer your sentences:
5) George hasn't played / hasn't been playing tennis for 2 months = both sound fine and can be explained with the repeated activity "rule" mentioned above.
6) The children are too tired because they haven't slept all night = here, I'd be more inclined to use the Present Perfect Simple and not the continuous. It seems the sentence was made the following day or at a later stage at any rate, so the period during which the kids should have slept is over, meaning the sentence is describing a completed activity, not a repetitive one or one which may still be in progress. I would however say, "The children haven't been sleeping lately" or "The children haven't been sleeping these days." I'd also consider "The children haven't been sleeping all night" fine if the night in question isn't over yet.
7) Lily hasn't spoken / hasn't been speaking to her sister since they had an argument = both are grammatically correct, though the Present Perfect Simple sounds better to me because it's more direct.