This is mainly a problem with imprecise wording, so we'll have to pick your statements apart very carefully:
small loops with high dI/dt on a PCB tend to produce predominantly magnetic waves and stubs with high dV/dt tend to produce predominantly electric waves in the near field. In the far field, both radiators result in EM waves.
so based on this you state:
1) I don't understand how in both cases (loop or stub) the resulting wave is not an EM wave (with a predominantly E or M component depending on the case)?
There's no such thing like an E- or M-Wave. Waves always consist of an electric and a magnetic field which periodically exchange energy. Maxwell's Equations describe that pretty unambigously: Change one over time and/or space, and you get the other. If you want a propagating wave, you'll get both. No way around that.
So, I don't know what your textbook actually says, but if it says "magnetic waves", get another textbook. I mean that. Pozar's Microwave engineering is the go-to textbook on that field.
If it says "changing magnetic field": yeah, a small coil is primarily an electromagnet. The "mechanism" with which energy is transferred from conductor to free space is electromagnetism. However, a changing magnetic field, per Maxwell, causes an electric field. No way around that.
2) I also don't understand how you determine how a specific structure will radiate (with respect to E field to M field proportion). Can someone please clarify?
That's antenna design. Basically, you can go the "old school antenna designer" ideology and see your structure as an impedance matcher between free space and conduction. Or you can just say "there's a current flowing in my structure, and I can describe that mathematically. Then I just throw Maxwell's equations at it, and see whether part of the energy leaves the system: radiation".
It's really down to reading a antenna or microwave engineering textbook in whole.
3) Why do these fields become an EM field in the far field? Is this because the free-space impedance (377ohm) forces any field traveling through it to become an EM field?
Free Space impedance is not a "force" that can "force something to do something". It's more of an effect. Please don't mix up cause and effect!
Maxwell's equations. Learn them. Live by them. They link changing E and H fields to each other, invariably. In some media, this leads to a fixed wave impedance. Wave impedance only simplifyingly represents relation between E- and H-fields in "easy" media. It's an effect, not a cause.
4) Why don't magnetic fields (ie rare earth magnet) or electric fields (ie electrostatically charged glass rods) become EM fields?
Maxwell. They don't change with time.
Since your profile says "Electrical Engineer", I trust you've heard of Maxwell's Equations before, or you'll have multiple lectures that revolve around them :) They are really the theoretical basis on which all of EE base their understanding of the world. (aside from the quantum guys. Those are special.)