For an accelerated charge to radiate, is an electromagnetic field as the source necessary? For an accelerated charge to radiate, must an electromagnetic field be the source of the force?
Would it radiate if accelerated by a gravitational field?
 A: The radiation, if considered classically, is independent for the reason Mark Eichenlaub gives. But considered quantum mechanically, it is not independent.
In short, photons are bosons. So the presence of radiation of a particular polarization and frequency will increase the probability of the particle radiating that polarization and frequency.
This is a topic I'd not seen before. I'll look around and see if I can find a reference to the effect.

An accelerated electron produces "synchrotron radiation". An example of the electromagnetic field altering the emission of such radiation would be "stimulated synchrotron emission". Stimulated emission was described by Einstein and is the physics behind lasers. An example paper combining these ideas:
Phys. Rev. Lett. 66, 2312–2315 (1991), J. L. Hirshfield and G. S. Park, Electron-beam cooling by stimulated synchrotron emission and absorption
http://prl.aps.org/abstract/PRL/v66/i18/p2312_1
A: I suspect that two charged objects orbiting one another due to gravitational attraction would radiate, but I can't support that assertion with a citation.
The question of whether or not a charge radiates when it is uniformly accelerated by gravity is an open question; read the link for an excellent discussion of why.
EDIT:
The link I posted isn't inspiring trust, so I searched for peer-reviewed work. I found two relevant papers:
Physical interpretation of the Schott energy of an accelerating point charge and the question of whether a uniformly accelerating charge radiates
The significance of the Schott energy for energy-momentum conservation of a radiating charge obeying the Lorentz–Abraham–Dirac equation
I'm not qualified to comment on the quality of the papers, but both attack the question 'does a falling charge radiate', implying an open question. I didn't find any experimental work on the subject.
This brings up a topic better suited for meta-discussion: I would like to see even more external citations in the answers here. I would also like to see more answers clearly indicate their logical foundation. Is your answer based on...
-  Original research?
-  Predicted by peer-reviewed theory but unverified?
-  Indirectly experimentally verified?
-  Directly experimentally verified?
Don't make us guess!
A: Your question is somewhat abstruse, but here's what I think you're asking:
Put a charged particle in a uniform external magnetic field.  The particle will move in a circular orbit, but since it's accelerating, it will radiate and its orbit will decay.
Now remove the magnetic field.  Grab the charge and forcibly swing it around in the same circle as before by some other, unknown means.  Does it still radiate in the same way as before?
The answer is yes because Maxwell's equations are linear.  Therefore we can analyze any situation in classical electromagnetism by superposition.
A: If you mean an external EMF, the answer is "The radiation is determined with an external filed". The charge acceleration is proportional to the external field, and a single  accelerated charge radiates.
If you mean the radiated field influence on the charge motion and subsequent radiation, the answer is "No" because the radiation is only expressed via external field. There is no need to invoke the proper field here.
In QM radiation of hard single photons happens discontinuously in time so QM is somewhat different.
A: According to the equivalence principle, a point-like charge in free fall will not radiate EM waves, because its movement its locally equivalent to any other inertial frame.
point-like masses will also not radiate gravitational waves in free-fall by the same reason; gravitational wave emission is a function of the third time derivative of the quadrupole mass moment; this basically means that only extended, non-symmetric masses will give such radiation (being extended means that significant parts of the mass will NOT be in a inertial free-fall frame)
Edit: The only other macroscopic force outside gravity (and the force that is not a force called exclusion principle) is the electromagnetic field. I don't think we have any evidence how accelerated charges radiate in other non-EM forces (i.e: weak and strong) but theoretically acceleration induced on hypotethical macroscopic fields of such should produce the same effect
