This discussion holds lots. Any way, In brief, mathematical descriptions in QM show that if we figure out the electron like subatomic particle we can understand the nature of atoms with lack of hundred percent assurance (I think a part of probability in QM regrades to that) .I didn't get what's you mean by 2nd question. experimentally we can see them due to the superposition of waves(wave packet)which make by elementary particles(this is the bizarre picture of nature!) .For more check:(Zeeman effect and or Stern-Gerlach experiments)
the number of electrons in an atom (if not ionized) corresponds to the number of positive charges (protons) in the nucleus. The number of negative charges can be counted. But you cannot distinguish the different charges. You cannot give them a fix number.
An example: The hydrogen molecule contains two electrons, but these electrons are undistinguishable. Both electrons exchange their states. You can ionize the molecule to get an electron, but it is impossible to say wether you have the left or the right electron.
To me, I prefer Rydberg experiment because the atomic properties are achieved from it based on spectrum.Meanwhile, spectrum corresponds to the cloud shells distributed around the nucleus (These shells basically made by wave-like electrons).