The temperature of an electron Does an electron have a temperature, if so, what is it?
Imagine an electron (Ke = 1 eV) in a tube at room temperature (300 K)

what is its temperature? 
Imagine now same electron in space (3 K) with same Ke, is it any different from the other?
What is the influence of external temperature on electrons?
Do those electrons have electric energy, or do we have such energy only when the electron hits something and discharges its Ke?
 A: Temperature is a measurement defined in the  mathematics of thermodynamics. Thermodynamic quantities emerge from statistical mechanics, so there exists a definition of temperature :

Except in the quantum regime at extremely low temperatures, the thermodynamic temperature of any bulk quantity of a substance (a statistically significant quantity of particles) is directly proportional to the mean average kinetic energy of a specific kind of particle motion known as translational motion. These simple movements in the three x, y, and z–axis dimensions of space means the particles move in the three spatial degrees of freedom. The temperature derived from this translational kinetic energy is sometimes referred to as kinetic temperature and is equal to the thermodynamic temperature over a very wide range of temperatures. Since there are three translational degrees of freedom (e.g., motion along the x, y, and z axes), the translational kinetic energy is related to the kinetic temperature by



where:
E_bar is the mean kinetic energy in joules (J) and is pronounced “E bar”
     k_B = 1.3806504(24)×10^−23 J/K is the Boltzmann constant and is pronounced “Kay sub bee”
     T_k is the kinetic temperature in kelvins (K) and is pronounced “Tee sub kay”

So  individual particles have kinetic energy, and temperature is defined by the average of the ensemble.
An electron within the hot element of the cathode ray tube, before emission will participate in defining the average temperature of the filament, when it is ejected it will have a specific momentum drawn from the kinetic energy distribution in the filament. In the vacuum of the tube, there is no ensemble to generate a temperature, the electron will keep this kinetic energy and increase it according to the field imposed that is attracting it to the cathode. 
By a hand waving  , if one were to assume that the kinetic energy of the single electron impinging on the cathode represents an average of an ensemble, one might say that the electron has the temperature of the sun plasma, for example, but it is a sloppy , not correct , assignment.  
A: Temperature is a measure of the average kinetic energy of molecules.
If there is thermal equilibrium the temperature is constant and on average the net kinetic energy gained or lost by a molecule during a collision is zero.
If a 1 eV electron is injected into a box which contains electrons which were at a temperature of 300 K what would happen?
Electrons at 300 K have an average kinetic energy of $\frac{1}{25} = 0.04$ eV and have a random in direction motion.
The 1 eV electron would collide with the other electrons and initially on average it would lose kinetic energy and the rest of the electrons would gain that kinetic energy.
In the process the motion of the 1 eV would be randomised and eventually it would become on average a 0.04 (plus a small amount) eV electron.
Injecting a 1 eV electron into a box of electrons at 7500 (= 25 $\times$ 300) K will result in the motion of the electron being randomised and with an average kinetic energy of 1 eV.
In deep space if the 1 eV electron "lives" long enough it will become a randomly moving "on average" 0.0004 eV electron.
A: The electrons of a CRT came from hot cathodes by thermionic emission. Typically the surfaces would be about 1000 K, and that would also be the temperature of the electrons. (But usually, there is no real thermodynamic equilibrium between the space charge and the solid.)
Then an electron gun would accelerate the electrons to energies of tens of keV, but that does not change the distribution of the electron velocities. It does not change the temperature of the beam.
