Okay, this might be a dumb question but I have to make it: could we essentially touch dark matter even though it is not electromagnetically interacting?

Dark matter can be sensitive to weak interactions, or even to new forces. It can also interact with the Higgs boson. Those are varied option to make it interacting (always super weakly) with the particle of the Standard Model.

So I get that they basically gather data regarding energy and momentum variations in the system and then create a virtual model? Correct me If I'm wrong. (I know you would :P).

It is actually different. One is colliding two particles along a so-called collision axis (that we may call the z-axis to fix the ideas). In the initial state, there is thus exactly zero momentum in the transverse plane (the Oxy plane) since our colliding particles do not move within this plane.

Now let us move on with the final state. ,Our detectors investigate what is going on transversally to the collision axis (i.e. in the Oxy plane) because the proton remnants pollute what is going on along the z-axis. From the detector records, one can reconstruct the final state and measure the total amount of transverse momentum in it (the amount of momentum along the Ox and oY directions).

By momentum conservation, we know in advance that this transverse momentum must vanish. If however often does not. Therefore, we calculate what is missing to make it vanishing. This is the missing momentum which we associate with dark matter in our case. The missing transverse energy is the norm of this vector.

Is it clearer?