What is the net charge (overall or sum of charges) of the nucleus and what number is it equal to?

The nucleus consists of protons and neutrons. Since neutrons are neutral, they are not taken account of in the net charge. The number of protons = the net charge of the nucleus and that is shown by the atomic number. For example, the atomic number of Oxygen is 8 which means that it has 8 protons; a net charge of + 8 only in the nucleus. On a deeper note, a neutral atom has the same amount of protons as electrons, so theoretically, oxygen is supposed to have 8 valence electrons, but sometimes it has a deficiency of 2 electrons (making an oxygen ion), so the net charge would be - 2 of the whole atom (since it's supposed to be neutral), but the net charge of the nucleus itself would still be + 8.