How many photons can an electron absorb and why?

Unboundedly many, because photon number is not conserved. Every time you push an electron with a classical field, you produce infinitely many soft-photons (if the universe is flat at infinity) and conversely, any long range field which pushes the electron has infinitely many soft-photons getting absorbed in a sense, although you can't tell photons apart, so you can't distinguish the ones that were absorbed from the ones that were emitted.

It would be useful if your profile gave an indication of educational background in physics or even of age.

I would recommend browsing through the CERN teaching resources.

How many photons can an electron absorb and why ?

Reminds of "how many angels can dance on the tip of a needle". :)

If you read the links provided you will understand that an elementary particle does not absorb a photon, it can interact with a photon and the result can be variable, but there will always be two particles in and two particles out, because of momentum conservation. The possible results of a photon interacting with an electron are drawn as Feynman diagrams. The same electron in its trajectory can interact with an unlimited number of photons.

Can all fundamental particles that can absorb photons absorb the same amount of photons and why ?

Particles interact, and do not absorb. And the interaction with photons will depend on the coupling constants in the Feynman diagrams. If a particle has no charge its probability of interacting with a photon is very low, through higher order diagrams, so no, not all particles interact with the same probability with a photon.

If we increase the velocity of a fundamental particle , do we then increase the amount of photons it can emit ?

A charged fundamental particle interacting with an electric or magnetic field can emit photons ( bremsstrahlung or synchrotron radiation). The higher the energy of the particle the higher the probability of a bremsstrahlung photon emited, so yes, the higher the energy the more photons from a charged particle accelerating in a magnetic or electric field. If the velocity is steady there is no emission.

At constant temperature and mass of an isolated fundamental particle that does not and will not move (constant speed and 0 vibration ) , is emitting a photon the only way to loose energy ?

Temperature has no meaning at the particle level. It is the kinetic energy of the particle in question. A particle can only loose energy via interactions with other particles/fields. Unless it interacts it does not lose any energy.

Edit for Clarification:

The above addresses the naive question on the number of photons an electron can absorb, and that is zero for free electrons and continuum photons: there is interaction and not absorption.

Electrons which are bound in atoms or molecules (or even crystals) are in a quantized state of energy. A photon with the appropriate energy can kick up to a higher quantized level the electron and then it will be absorbed/disappear. In this case, of a potential well, one photon can be absorbed by the system electron-in-potential-well at a time. There could be a second appropriate energy photon which could kick it up again but the times this can happen are countable, and finally the electron will be free and the atom ionized. Usually the electron cascades down to the lower level emitting maybe more photons of lower energy as it falls. A specific bound electron can help in the absorption of a photon by the system a limited number of times.