Can a charged particle interact with its its own electromagnetic field? Say an electron begin moving with velocity V back and forth very quick or at the speed of light, could it interact with its own previous pre existing electromagnetic field?
 A: A charge always interacts with its own field. This is one if the causes of infinities but also of radiative corrections in the theory of quantum electrodynamics. 
A: First, an electron does have a rest mass, and cannot move at the speed of light.
The static electric field around the electron is mediated by virtual photons. Virtual photons are off mass shell, and do not have to obey all laws of physics. For example, they do not have to obey the speed c limit. 
There are two cases, when the electron moves at constant speed or accelerates:


*

*No acceleration


In your case, when the electron moves with a speed close to c, the static electric field around the electron is still moving with the electron.
What you are saying is that the electron would be moving faster then its own static electric field. That is not possible. The electric field is mediated by virtual photons, which can propagate theoretically even faster then light, they do not have to obey the speed c limit. These virtual photons are just a mathematical method to explain the static electric field around the electron, they are not real particles.
There is no consensus on this site about whether the electric field around the electron is moving (interacting) faster then c or at the speed c. For example when you turn on an electric charge, the static electric field around it extends at speed c, so not instantaneously. 
But let's disregard the fact that virtual photons are just a mathematical theory for describing the electric field and these virtual photons do not have to obey some laws of physics, like the speed limit c, and let's say that the electric field can interact at the speed c.
Even if the electron would be moving very close to speed c, from the viewpoint of the electron, the static electric field around it is  would still be moving at speed c. As per SR, from any inertial reference frame, like the moving electron (not accelerating), the electric field seems to be moving at speed c. So however fast the electron is moving, it cannot interact with its own field (previous pre-existing). This is because the electric field is co-moving with the electron. And the electron has rest mass, is moving slower then c, hass an inertial reference frame, and from this reference frame, the electric field seems to be propagating at speed c. 
So the previous electric field that the electron created at the previous position in space, has already co-moved with the electron, so the electron cannot self interact with it.
You are asking if there are leftovers of electric fields that the electron created when it was at a previous position in space. The answer is that there are no such previous fields, or their leftovers. There is only one field (we are talking about the near field), that co-moves with the electron. 
So if you are asking about the near field, the electron cannot self interact with its own previous electric field.


*Acceleration


When the electron accelerates, it does not have an inertial reference frame any more. What happens, is that the electron starts to interact with the field, it stirs the field up (its near field and the far field too), and create radiation. That is EM waves, photons, it emits photons. So when you are asking whether the electron can self interact with its own previous near field, yes, if the electron is accelerating.
