EEG, PSI interpretation
Can one guess what stage of experiment every matrix refers to? If so, how? I mean, do they at least look like the result of the experiment described? If they do, how can one interpret them?
  


I do not think that I can. But here is a guess that might give you some insight: The most "populated" matrices are those for a single hand up in the air (whether left or right). 

Why?

Functional connectivity metrics (of which there are so many by now) assess some of "dependence" or "correlation". There is a higher chance for any two pairs of waveforms to have more correlated signals when the diversity of the generated signals is low. With two hands being raised, the brain generates a large set of impulses to drive all the necessary muscles. Consequently, the chance of them having a high correlation is low.

If you notice, a key characteristic of functional connectivity estimation in related papers is some form of averaging, whether temporal, spatial or across and within subjects.

Whether it is Magnetoencephalography (higher spatial resolution than Electroencephalography), or of course, fMRI or EEG, there is an intermediate stage that "summarises" observations.

In fMRI you sum voxel traces around anatomical areas indicated by an atlas, in MEG you sum contributions of elements around broad areas (simplest method) OR you do some form of source localisation and then look at the connectivity of the localised sources. Similar workflow applies in EEG too (e.g. LORETTA as above).

This is a required step to increase the Signal to Noise Ratio (SNR) and look at (hopefully) robust correlations. Along this idea are also methods that examine functional connectivity in specific bands of brain activity

Therefore, I doubt that you will get anything meaningful by looking at each individual channel without some form of spatial or temporal filtering / averaging. 

This however, is half of the story. The other thing you can do is, of course, to change the task.


  
  What could be an experiment that would show the noticeable PSI for two areas given beforehand? I mean, we had two areas given, pre-motor cortex and primary motor cortex, and we supposed that we would find a synchronization between them in such a motion experiment; something's gone wrong, though. If the problem is in the design of experiment, what design would be better?
  


You will definitely see differences in the default mode network between two radically different tasks. For example, at rest and at "work".

To reduce your post-processing workload, try to use tasks that are not likely to generate a lot of artifacts. For example: Resting state and "solve a mathematical problem".

This design will give you something to compare "activity" against and help in reducing spurious correlations further.

Finally, be aware of filtering. The more your filter a waveform, the more it "looks like the filter" and therefore, the more chances there are to have more increased correlations.

Hope this helps.