Does Chemical Formula of Matter Influence on how would it be Charged?
Firstly: Suppose we rub glass ($\ce{SiO2}$) against paper (which is made from cellulose $\ce{(C6H10O5)_{$n$}}$). We receive positively charged glass and negatively charged paper. Is that because of total ionization energy of molecule stronger than glass or crystal system too?
  


Although atoms and molecules like to be uncharged there is a difference between the electron affinity and the ionization energy. Electron affinity is a measure of willing a substance is to accept an extra electron. Ionization energy is a measure of how willing a substance is to donate electrons. 

When the paper is rubbed against the glass the surfaces of the two substances come into physical contact. So the electron affinity of the paper is greater than the ionization energy of the glass rod, so electrons are stripped from the glass and are transferred to the paper. Enough electrons are transferred to balance the affinity/ionization relationship. This is known as the Triboelectric effect.  

To be totally honest the Triboelectric effect is not well understood. There are a lot of enigmatic interactions in how the surfaces contact, how many times the surfaces contact when rubbing, how specific contact points my heat up, or how some physical transfer may take place. Read the Wikipedia article for more details on these points. 


  
    Secondly: When we rub atoms against other atoms, do electrons accelerate each other by the same electrostatic charge and change their orbitals to higher orbits due to synchrotron radiation they emit. Or is it because of the fact that protons are more close are ionizing electrons by electrostatic attraction? Am I correct?
  


There are a number of erroneous concepts in the setup/question. The first explains the phenomena.

Synchrotron radiation is electromagnetic radiation and has absolutely nothing to do with the electrostatic experiment that you are discussing. 

The electrostatic experiment that you are discussing does not "accelerate electrons." So rubbing a glass rod with paper doesn't make a particle accelerator. 

Neither the paper nor the glass has bare protons floating around.