You've got quite a few questions here all in one, but will try to touch on all of them:
The "identity of an atom" is an agreed upon definition: the identity is defined by the number of protons. So loosing or gaining electrons will not change its identity. However gaining and loosing electrons can and does change the properties and behavior of an atom.
Not sure what you mean by "unstable" or what you are referring to when you say that "an atom with not enough electrons becomes unstable". In general, all ions (atoms with either more, or less, electrons than protons) tend to become more chemically reactive. That is not necessarily the same as unstable. Unstable can also mean that it may be difficult to maintain certain ionic state: that is, some ions will have a strong tendency to either gain or loose electrons until they become either a uncharged atom, or an ion that is more stable. But in this context we are talking about the stability of the atom's electronic structure, not of the nucleus. Thus there is not (under most normal circumstances) an issue of gaining or loosing protons simply because the atom has gained or lost electrons.
Regarding your last question "Can you have too many electrons in an atom?" The answer is no. As you add electrons to an atom, a point will be reached where that atom will not be able to accept more electrons (as such a process is energetically unfavorable). This gets back to the "stability" question, but again we are talking about the stability of the atom's electronic structure (not its nucleus). If an electron somehow manages to have enough energy to get into the electronic structure of an already highly negative atom (ion), that new, more negative, ionic state may be unstable: but what this means is that the atom would very likely, quickly loose the additional electron and return to a more stable, more favorable energy state.
Finally, regarding the issue discussed in the other post to which you provided a link, i.e. whether positive ions with no electrons are considered atoms, it really depends on the context of the discussion. A hydrogen ion, H+, is also just a proton, but typically in a chemical or biochemical context we regard it as an ionized hydrogen atom. Again, ions have different behaviors than their uncharged atoms, but we still typically "identify" them by the number of protons in our discussions.
Regarding your comment on the other answer: "... where do you go to ... find answers? So how do you know what the fixed capacity of electrons for a atom is? How do you know it can't accept 6 electrons?"
The questions that you are asking are for the most part Basic Chemistry. Get a good basic chemistry textbook and read the first few chapters. There you will learn about the electronic structure of atoms, orbitals, how many electrons each orbital can hold, how much energy it takes to add or remove electrons. There is also a concept called "electronegativity" which you should pay attention to. It is a measure of how strongly a given atom's nucleus holds onto the electrons.
Questions of stability, and how many electrons can be added or removed always come down to energy. If an atom is highly electronegative (has a very strong hold on its electrons) then it may take a lot of energy to remove an electron and make a positive ion. And each electron removed may increase the atom's electronegativity so that removing the next electron takes even more energy. This is especially true for a "full" electron shell configuration (again learn about this in a basic chemistry textbook: full orbitals or electron shells are very energetically stable, and therefore it takes a lot of energy to either add or remove an electron). The amount of energy required will ultimately (in practical terms) limit the number of electrons that can be added or removed.
Aside from a good basic chemistry textbook, here's one place where you can get some basic concepts without too much detail:
Start here: Elements and Atoms
Then here: Electron Shells and Orbitals
After the above, if you want a lot more detail, get a good textbook or try these: