A typical highly excited atom has an outer electron that sees a charge +1 remainder far away, essentially at a point, and such a configuration is indistinguishable from a Hydrogen atom, in terms of the transitions of the outer electron. Instead of orbiting the proton, it orbits the core of remaining electrons.
The hydrogen atom in a zero temperature environment has infinitely many states, indexed by n,l,m, and the spin of the electron, and the energy goes as $R/n^2$ where $R=13.6eV$. Thermal energies at room temperature are 1/30 eV, so if the atom is excited to the n=20 state or above, it will ionize with thermal energy, and this is the limit of n. Beyond this limit, the Hydrogen atom at room temperature will not stay together.
For an effective hydrogen atom in these configurations, where one electron is orbiting the rest, the limit on n is about the same as at room temperature, except the asymptotic approximate-hydrogen "n" is not the same "n" as the n appropriate to the nucleus you are considering. At a temperature of 4K, you can go to n=100 before you ionize the atom.
There are restrictions to how a single photon can excite an atom, which are called the selection rules. These follow from the fact that the X operator in quantum mechanics has a lot of matrix elements which are exactly zero. Absorbing a photon can only link states with orbital angular momentum l different by at most 1 unit, and "m" different by at most one unit.
In addition, there are also qualitative selection rules, because the matrix elements for the X operator decay away from the diagonal. They decay quickly for circular orbits (l the same size as n), which are classically differentiable, and they decay slowly away from the diagonal for highly elliptical orbits (l much much smaller than n), depending on the orbit. So to excite an atom from a circular orbit (large l relative to n) to a much higher n with the l in the range of the selection rule is practically difficult. But l=0 and l=1 states at high n are highly elliptical orbits, which are classically singular (l=0) or nearly singular (l=1), and these have powerlaw decaying X matrix elements to different states at large n. These things are studied in the experimental field of Rydberg spectroscopy.
The precise answers:
- You can theoretically excite an atom to very high levels, but there are restrictions on how you can do it using dipole absorption of a single photon, you have to keep knocking it up in energy gradually, through several intermediate states. The high energy levels correspond well to classical motions, and include integrable motions where one electron is far from everything, and solar-system states, where several eletrons are orbiting at very different distances, and chaotic many body states, where a few high-n electrons are interacting strongly. At zero temperature, there are always infinitely many levels
- No, the energy emitted in different atoms is not the same, not even close. One difference is that the nuclear charge is bigger, so that the 3p state is a 3p state of a different nucleus, but there is also the fact that electrons interact, so adding more electrons shifts all the level energies profoundly. The energies of the outermost electrons in atoms which fill up to n=3 (those on the third line of the periodic table) is more like the energies of the outermost electrons in atoms which fill up to n=2, this is Mendeleev's correspondences.
It isn't even true that the same atom with the same state will emit exactly the same wavelength each time it is knocked into an excited state. The range of frequency the atom emits is called the natural line-width, and it is the reciprocal of the lifetime of the state.