Why accelerating charged particles radiate EM waves? Why do they do it? 
If I apply some constant force on a charged particle, would it gain velocity just like any other particle or would it lose its energy by emitting radiation?
What if I observe a charged particle from an accelerating frame, would the particle still radiate energy? If so, where does that energy come from ? 
Please explain clearly about how the velocity of an electron increases when a constant force is applied.
Not a duplicate of How and why do accelerating charges radiate electromagnetic radiation? because my questions are not answered there.
 A: That light was electromagnetic radiation was established after Maxwell's equations,ME,  were developed. Before that, electricity and magnetism and light were three different stories. The equations allowed us to explore interactions between charges, magnetic fields and light , the predictions fitting the data and validating the classical electromagnetic theory with no exceptions, except when reaching quantum mechanical microstates.
You must have noticed that the ME are dependent directly on the electricity and magnetism laws discovered experimentally before the unification into ME.
Now the question "why" does not apply to laws. The only answer is "because that is what has been experimentally observed", period. MEs , directly based on these laws, are also of the same validity, i.e. they exist because they are based on experimental observations and the mathematics works. So the simplest answer on 

If I apply some constant force on a charged particle, would it gain velocity just like any other particle or would it lose its energy by emitting radiation?

Is that it would lose partially its energy, depending on the inertial system and the acceleration of this, because that is what the mathematics of ME predict, and it has been observed experimentally, after the prediction, validating ME.

What if I observe a charged particle from an accelerating frame, would the particle still radiate energy? If so, where does that energy come from ?

You are always in your center of mass system. To accelerate, energy has to be supplied , and it is supplied by the accelerating agency. Depending on the system under observation, there will be in your center of mass system radiation observed to come from the charged mass, the energy supplied by the system accelerating you. This will need calculations using ME and the boundary conditions of the system, i.e. sizes, distances etc.

Please explain clearly about how the velocity of an electron increases when a constant force is applied.

Electrons are elementary particles in the quantum mechanical regime. They are accelerated in beams, and it is not something simple. See this link for a description. The reason why linear accelerators  are used is because the electrons accelerated in straight paths radiate much less than in circular accelerators, and thus linear accelerators are energy efficient. Again this is a subject dependent on ME and the Lorenz transformations inherent in them. It depends on the mathematics and cannot be handwaved, except that the predictions of the ME work in the lab. It is a whole field of study and construction that uses the results.
A: 
If I apply some constant force on a charged particle, 

... you do this by the help of an electromagnetic interaction or a magnetic field. Electromagnetic interaction is somehow like the exchange of energy for billiard balls, only instead of balls the involved objects are electrons (as well as the other subatomic particles) and the exchange happens via photons. And this photons are real, not virtual one. So throwing a ball with your hand, in detail, the electrons on the surface of your hand interact with the electrons on the surface of the ball. The first lose energy, the second gain energy.

would it gain velocity just like any other particle or would it lose its energy by emitting radiation?

In any interaction the system with the higher energy content lose energy and the system with the lower energy content gain energy.
It is in the nature of the matter that this exchange is accompanied by energy losses, again in the form of photons. Remember the second law of electrodynamics:


In a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases


, so whatever you do, as a secondary process you produce photons.
But what is about the influence of a time varying magnetic field on a charged particle or of a magnetic field on a moving particle? The magnetic field deflect the particle and from our observations we know that any acceleration - and the deflection is an acceleration - is accompanied by the emission of photons. The charge, emitting photons, lose energy. It’s kinetic energy decreases and the defections from the magnetic field together with the energy decrease let the particle move in a spiral path until it come to rest in the center of the spiral path.

What if I observe a charged particle from an accelerating frame, would the particle still radiate energy? If so, where does that energy come from ?

It would not radiate, because underwise the lose of energy would break the particle and this is not what we experience. If you feel an acceleration, you are in an accelerating frame and you will radiate.
