Can a free particle ever emit a particle? I have found this question here Can a free particle absorb/emit photons?, along with other resources that show a free particle cannot emit a photon (in a vacuum).  
Now, I am 90% sure it does, but does this result hold generally, ie not just photons/ photons but in a medium?  
If you go in the rest frame no matter what it emits energy is not conserved.  The reason I am questioning this is that an exam question said 'if it can emit it find the angle relative to the motion of the electron'.  Seems odd that they would give away marks for just repeating that it can't.
 A: True and true - there is no contradiction here.
The complete answer is contained in the answer to linked question:

This is because energy and momentum are not both conserved if a free
  charged particle (say, an electron) emits a photon

and the comment to that answer:

Note that if the particle has internal structure, this argument can
  fail. For example, atomic nuclei emit gamma rays. This is because they
  have more than one internal state with different energies.

Applying these two principles we arrive at:


*

*A single free electron cannot emit a photon

*A single free hydrogen atom with electron in an excited state can emit a photon by spontaneous emission

*A single free nucleus can emit a photon again by relaxation of its internal structure.


Note: In this answer I have only considered the dynamical restrictions involved in emitting a single photon. As pointed out in the comments there are many more complex decay processes of either elementary (e.g. tau) or composite (e.g. neutron) particles which may result in the emission of a photon as one of many decay products.
A: 
If you go in the rest frame no matter what it emits energy is not conserved.

But note that this argument fails if the parent particle is massless, since then there is no such rest frame. A gluon can decay into two gluons. See Decay of massless particles .
A: The answer may be found on Wikipedia's page
Electron scattering
in the explanation of Synchrotron emission:
"If a charged particle such as an electron is accelerated – this can be acceleration in a straight line or motion in a curved path – electromagnetic radiation is emitted by the particle. (...)
Robert Langmuir is credited as recognizing it as synchrotron radiation or, as he called it, "Schwinger radiation" after Julian Schwinger. (...)
Within a circular orbit such as a storage ring, the non-relativistic case is simply the centripetal acceleration."
Different from Bremsstrahlung where electrons are defracted by the electromagnetic-field of a particle which in that case is an atom,  "Schwinger radiation" shows that there are no defracting "third" particles needed for an electron to emit a particle which is a photon.
A free electron can emit a photon simply by being accelerated.
