Is your first question is asking why the proton count determines the element as opposed to some other property?
I suppose the reason why is because it has the most significant physical effect of all the atomic particles. Neutrons don’t have charge, so they cannot affect other particles electromagnetically. Electrons do, but the nucleus of an atom can easily pick up or lose electrons while a nucleus will not lose or gain protons under normal circumstances. Therefore it’s more convenient to categorize elements by the number of protons rather than the number of electrons.
As for how this number translates into the properties we observe, it has to do with the way elements bond with each other in chemical reactions and the resulting geometry of the atoms due to these bonds. For example, graphite, AKA pencil lead, which is a form of carbon, leaves marks on surfaces because the atomic structure of carbon resembles a series of strongly held carbon atoms arranged in flat planes, each plane held together by weaker bonds. When you draw with a pencil, you are essentially removing these weakly bonded planes of carbon.