Can atomic number change the type of electromagnetic radiation being emitted when a electron drop from a energy level to another? Without getting into too much/any math, I just wanna talk about intuitions if possible. 

If atomic increase, the attraction increase, and consequently the potential energy increase. More energy leads to higher frequency, different radiations. 



I guess the reason if this diagram is universal is that the gap between 1 and 2 is too large too that the difference in energy cannot simply be compensated by adding more protons.
Absolutely, the differences in potential energies will be different for every element. This is because the potential energies will be different for every element. Notice that the potential energy for each energy level can be determined from the ground state. Let $E_g$ be the potential energy of the ground state (with the sign). Then for energy level $n$ the potential energy is $\frac{E_g}{n^2}$. Thus Differences between energy levels $n$ and $m$ can be written as $E_g\left|\frac{1}{n^2}-\frac{1}{m^2}\right|$.

Clearly it will be different for a different ground state energy which is precisely what the number of protons affects.

Note that the only important part of the mathematics is that all differences rely on the ground state potentials.