Do hydrogen atoms have inner and outer electron shells? I saw a diagram somewhere that showed a carbon atom with an inner and outer shell, is this proven? Are these shells magnetic or something else? Do you consider the outside of these shells as the diameter of the atom?
Hydrogen only has one electron, so it makes no sense to talk about shells. As for other atoms, different electrons occupy different states (atomic orbitals), which can be classified according to shells. Only the outermost electrons, called valence electrons, are relevant in most cases (spectroscopy, chemistry, etc.) Usually, the word shell is applied to all electrons who share the same principle quantum number ##n##. Because of quantum mechanics, the actual shape of an atom is "fuzzy". For an isolated atom, you could use something like the sphere within which there is a 90% probability of finding all electrons. For atoms bound into molecules and solids, equilibrium bond distance can be used as a starting point.
What keeps the atoms from intermingling electrons. If hydrogen atoms were pressurized in a vessel, what would keep them from mixing together? What is the official border of a H atom? What is the average diameter of an H atom?
They are intermingled, in the sense that the electrons are not on neat orbits, but occupy diffuse orbitals. For instance, there is a non-zero probability of finding even a valence (outer) electron at the nucleus. But the rules of quantum mechanics (the Pauli exclusion principle, to be precise) make it such that two electrons in teh same atom cannot be in the same state, and therefore the more electrons an atoms has, the greater the number of states (orbitals) needed, and these increase in energy, such that electrons are found on average farther and farther away from the nucleus. At one point, two atoms will start exchanging their electrons and you will form a molecule There is no "official" border, the calculation always has a part of arbitrary in it. In the Bohr model of the atom, with is semi-classical (meaning not fully quantum mechanical), the electron orbits at a radius of ##0.52 \times 10^{-10}\ \mathrm{m}##, so that gives you an idea of the size of an atom.
I completely agree with everything DrClaude said. I'd just like to somewhat rephrase this, because it might come off wrong: It is perfectly possible to calculate (very accurately) the average electron density distribution in an atom, or any statistical moments of it. For example, one can evaluate the expectation values <|r|^n>, where r is the operator of electron distance from the nucleus. The arbitrariness now only comes from assigning atomic borders based on that. For example, one could define something like √(<r^2>) or <|r|> as the distance (both being equally valid), or the distance r where there is less than 10% (or 1% or 0.1%...) of finding an electron at radii > r. Technically every electron of every atom has non-zero expectation values of being anywhere in space, and where you want to consider this probability as ``small enough'' is more or less up to you. The situation is even more complicated when it comes to talking about atoms in molecules, because technically the atoms lose their identity in molecules andcannot be strictly separated (but that's not the entire truth, either, because often they, in fact, can).