Should experiments of high energy electron collisions be done as an experimental test of the Standard Model? As I see from this discussion link there were no experiments where high energy electron beams were collided, only at low energy (VEP-1 collider in Novosibirsk, Russia). Assuming that standard model (SM) is correct we should see generation of electron-positron pairs in electron-electron-collisions when energies of the colliding electrons are high enough. So, the question is following. Should such experiments be done to test SM since if we will not see the electron-positron pairs the validity of SM is under a question.
 A: Quite contrary to the answer mentioned in the link you provided, there were electron electron collision experiments. In fact, they were the first collider experiments that have been conducted.
So, electron electron collisions are also called Moller scattering which is well described quantitatively by the feynman rules for Quantum Electro Dynamics.
First, there is VEP-1 collider in Novosibirsk, Russia. It is a collider with an energy of $2*160 \,\mathrm{MeV}$. It reached a luminosity of $4*10^{28} \,\mathrm{cm^{-2}s^{-1}}$. The experimental studies on scattering of electrons were conducted and the following reactions were observed
$$e^{-} e^{-} \rightarrow e^{-} e^{-} \gamma$$ 
$$e^{-} e^{-} \rightarrow e^{-} e^{-} 2\gamma \tag{Double bremstrahlung}$$
So, the first reaction can be seen as a pair annihilation producing a photon. But, most probably it was from bremsstrahlung of either the initial or final states.
Secondly, there is also a collider in the US which is the Princeton-Stanford Experiment Collider. It was the most powerful electron accelerator during it's time. It is a collider with an energy of $2*500 \,\mathrm{MeV}$. It reached a luminosity of $2*10^{28} \,\mathrm{cm^{-2}s^{-1}}$. It has conducted several tests for QED.
QED is very well tested and understood. So, according to me we do not require a electron electron collider even at high energies today. I am more excited to see a Linear Collider of electron positron of higher energies or a muon anti-muon collider.
Also, if you would like to read about these colliders here is the link: https://arxiv.org/abs/1307.3116 
