I would say that one experiment that demonstrates the atomic nature of things is the observation of Brownian motion. But it is not the experiment itself that convinces that things are made of atoms, rather its theoretical explanation given by Einstein in one of his 1905 papers (actually Einsteins work for his PhD was on the subject of atomic theory and there are several publications in the period 1903-1905). Of course there is also the observation of Rayleigh who calculated Avogadro’s number by the distance from which he could make out the figure of Mount Everest, assuming that light is scattered by atoms and that is why far away objects look fuzzy (1,2). Also scattering experiments demonstrated the atomic nature of things.
(1) Rayleigh, On the transmission of light through an atmosphere containing small
particles in suspension, in Scientific Papers by Lord Rayleigh Vol. 4, pp. 247–405,
New York: Dover, 1899/1964.
(2) P. Pesic, Eur. J. Phys. 26, 183 (2005).
(3) Patterson, G. Jean Perrin and the triumph of the atomic doctrine (2007) Endeavour, 31 (2), pp. 50-53.
The divide is actually not between covalent and ionic, but rather a spectrum between localised and delocalised electrons. The history of all this is actually quite fascinating, and Phil Anderson in his book "More and Different" has a nice chapter on this. Essentially, around the time that people started doing quantum mechanics on molecules seriously, there were two schools of thinking which dominated.
On one side was Mott and more popularly, Hund and Pauli who thought of electrons as primarily attached to atoms and through electromagnetic interactions their motions/orbitals would be deformed and one gets molecules. This is the version usually taught in chemistry classes as with a few rules of thumb it is possible to qualitatively account for a vast range of behaviours.
On the other side was Slater with a dream of a machine which could simply compute the electronic structure by giving it the atoms and electrons. In this picture, the electrons are primarily thought of as delocalised over all the atoms, and through a rigorous procedure of perturbation theory one adds the effect of interactions between electrons and may achieve arbitrarily good precision.
The latter has the problem that the results are not intuitive --- there are no rules of thumb available and one is reduced to simply computing. The problem with the former is that to achieve high accuracy, the "rules of thumb" become exceedingly complex and are not really very easy to use or to compute with --- it lacks the simple regularity of the Slater dream machine. It is telling that essentially the latter has won, and nowadays it is routine to compute the electronic structure of quite large molecules (~1000 atoms) through brute-force (the technique is known as density functional theory, and there are commercial software available to do it).
In finite molecules one can actually show that in principle both approaches will work --- technically we speak of there being an adiabatic connection between the localised and delocalised states. The only practical difference is just how hard it is to carry out the calculations. However, in infinite molecules (e.g. solid crystals) this is not true, and there can be a proper phase transition between the two starting points. In that case, the localised approach corresponds to what is fancily called these days "strongly correlated systems" such as Mott insulators and magnetically ordered materials, and the delocalised approach are essentially metals (technical language: renormalises to be a Fermi liquid).
Nowadays there is a desire (from theoretical condensed matter physicists) to develop the localised approach again, as it may be possible to find some useful rules of thumb regarding magnetic materials, a prominent example of which are the high temperature superconductors.
Best Answer
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.
P.S. 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:
Electronic Structure of Atoms: Introduction
Electron Configurations