How do they experimentally find the rest mass/energy of an electron? Can you explain how can the energy of the rest mass of an electron be determined with precision? Is it the heat of the radiation of a pair-annihilation that is measured?
Also, how does the content of KE of the particles affect the value of the energy which is measured?
 A: The most accurate measurement of the mass of the electron  to date  is not in accelerator experiments. The error margins  are large in the eletromagnetic calorimeters. 
From the abstract of the 2014 paper:

Here we combine a very precise measurement of the magnetic moment of a single electron bound to a carbon nucleus with a state-of-the-art calculation in the framework of bound-state quantum electrodynamics. The precision of the resulting value for the atomic mass of the electron surpasses the current literature value of the Committee on Data for Science and Technology (CODATA6) by a factor of 13. This result lays the foundation for future fundamental physics experimentsand precision tests of the Standard Mode

Italics mine
from the paper:

The  key  tool  for  our measurements  is  the  Penning  trap .

QED precise calculations are involved in extracting an accurate value. 
A: One simply measures the energy of the photons (most often exactly two, though three is frequent enough to be worth discussing and high multiplicities happen) resulting from electron-positron annihilation. You can use a lot of different detectors for this, but to impress the layman I'd use a high purity germanium detector for the precise line-width, though a NaI crustal has a more linear response, and liquid scintillator offers an easy means of building a $4\pi$ calorimeter.
Do it in a collider setting and the center-of-mass frame kinetic energy is added to the total.
A: If we already know the electric charge of the electron $q$, its mass $m$ can be derived from measured frequency of radiation $f$ it produces when in circular motion in uniform magnetic field of known intensity $B$ (so-called cyclotron frequency). This frequency obeys the relation
$$
f = \frac{B}{2\pi}\frac{q}{m}.
$$
https://en.wikipedia.org/wiki/Cyclotron_resonance
