Terminology: Is there a good way of saying "non-quantum mechanics" physics? Does a term for non-QM physics exist - in the sense of classical physics including relativity?
 A: The correct, modern, professional terminology for "non-quantum mechanics" is "classical mechanics". 
Students are routinely confused by this because there is an old-fashioned tendency in texts that discuss special relativity to use "classical" to mean "non-relativistic". That usage only exists in educational genre materials, and is not used professionally. (However, because it's part of every professional's education, you do sometimes hear professionals using "classical" this way when speaking casually.) 
Here in the 21st century, the correct terminology for "non-relativistic" is in fact "non-relativistic", not "classical". So, you hear phrases like "non-relativistic quantum mechanics" to describe the subject commonly discussed in undergraduate quantum mechanics courses, or "relativistic quantum mechanics" for the theory that doesn't quite work out and forces you into quantum field theory. You can talk about "relativistic classical mechanics". That's totally fine and a real thing professionals say, although many students think its an oxymoron. The topic of most introductory physics courses is "non-relativistic classical mechanics", which is routinely shortened to just "classical mechanics", or just as often "Newtonian mechanics", which leaves no ambiguity about whether the classical mechanics is relativistic or not. 
A: The way I use it, "classical physics " includes classical mechanics, classical electrodynamics, special relativity and general relativity.
Otherwise I use classical mechanics, classical electrodynamics ....
Quantum physics for me includes quantum electrodynamics and quantum field theory.
I do not think there is  general terminology though.
A: I've heard the term Newtonian Mechanics and Newtonian Physics. And also 'Jackson E&M' where Jackson was author of an old E&M text for classical (using Anna's answer) E&M.
A: I don't think non-QM is used. At least I have never heard of anyone using it. It is always classical physics when you mean anything non-quantum mechanical. 
