There are a lot of questions, so I assigned a number to each one.
Let's assume we have electron which is not moving in some inertial frame of reference. Let this electron be hit by a photon. Since photon has a momentum, the electron start's to move with constant velocity v. Thus it changed its velocity from 0 to v over some time t, so this would imply acceleration.
Is it true that during such process there is some amount of time, for which the electron has acceleration, or rather such "time of interaction" is singular/infinitesimal(1)? If the latter is true, then wouldn't that mean that speeding up particle is non continuous process(2)?
But if speeding up is a continuous process, then since electron is accelerated for some amount of time, can it spontaneously emit the photon back to the universe during that time interval(3)? Would that mean it could bounce the whole photon back, or deflect it, or emit photon with smaller energy in some direction, and start moving in yet another direction(4)?
Can it even emit a photon with higher energy than it initially absorbed(5)? For that one we can assume that earlier our electron absorbed the photon and started to move, and then it was hit by another photon, which would give the electron an "acceleration window", which would allow emit back both photons(or one with the combined energy).
If in the above scenario, the answer is affirmative, then would it be still affirmative, if the electron would be hit by the first photon, then after it got constant velocity, we would change frame of reference to such the electron is not moving, and then hit it with a second electron(6)?
Note that if the answer for question (2) is affirmative, then questions (3), (4), (5) and (6) do not make sense anymore.