Can electrons emit light when they are independent? If an electron is independent, it is an individual electron, i.e. not in the orbit of the atom. Is it able to emit light or does the orbit have to do with the production of light?
 A: Accelerating and decelerating  charges produce electromagnetic radiation.
Bremsstrahlung :

Bremsstrahlung (German pronunciation: [ˈbʁɛmsˌʃtʁaːlʊŋ] ( listen), from bremsen "to brake" and Strahlung "radiation", i.e. "braking radiation" or "deceleration radiation") is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into a photon, thus satisfying the law of conservation of energy.

and synchrotron radiation

The electromagnetic radiation emitted when charged particles are accelerated radially (a ⊥ v) is called synchrotron radiation. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers

also cyclotron radiation

Cyclotron radiation is electromagnetic radiation emitted by accelerating charged particles deflected by a magnetic field

All these radiations have a continuous spectrum, in contrast to the radiation coming from atomic energy level transitions: electrons falling from an excited level to a lower energy level and a photon of specific energy  emitted.
A: Free electrons can produce electromagnetic radiation i.e light, if they are made to oscillate. The oscillating electric field produced by the electron will create a perpendicularly oscillating magnetic field which will combine to produce light with the frequency of vibration of the electron. An electron returning to a lower energy level from an excited state also produces light in order to lose the "extra" energy it had in its excited state. But that is not relevant to the way a vibrating free electron can produce light.
A: No, it would violate both Conservation of Energy and Conversation of Momentum, this is without using Quantum Theory.  A third law is Conversation of Angular Momentum, an electron has spin of 1/2 while a photon has spin of 1.
Electron and positron can annhilate producing 2 photons if their spins cancel, or 3 if they are parallel.  In a Centre of Mass frame it is easy to see how energy and momentum are conserved.
