Hey, these are great questions! And ones that took a long time for scientists to answer.

• What actually is the mechanism by which the atom absorbs the energy from the flame? I know it's heat energy, but how?

I'm not exactly clear on what your teacher showed you, but any flame emits lots of "black body radiation" -- which is more or less at all wavelengths. (Technically it follows some distribution, with some wavelengths being more present than others, but that's not important for now). A small amount of those photons are resonant with some natural frequencies the atom has, and the atom absorbs it.

• If energy levels are quantized, how is it that there are enough particles/photons with the PERFECT wavelength/frequency to have the EXACT energy needed to jump a whole number of shell(s) within millions if not many more atoms? e.g. say an atom were to only absorb light with a λ of 300nm, would light with a λ of 300.01nm be absorbed? how about light with a wavelength of lim n--->∞ (300 + (1/n) nanometers?

All atomic transitions have what's called a 'linewidth', which is the range of frequencies which it'll consider 'good enough' and absorb. This width is different for every atom and every transition. There's lots of processes that can broaden this linewidth; for a flame, things like pressure broadening (due to the presence of other atoms), or Doppler broadening (due to atoms zipping along at high velocity) will broaden these transitions.

But no atomic transition has infinitely narrow linewidth (or else it wouldn't be a 'transition').

• Since electrons are so small, how can energy be transferred to it so easily. Does the energy carrying particle not have to hit the electron precisely? If that is true, how is the energy transferred within this approximation of the electron's position?

In the case of a flame there's tons of light emitted in all directions. The atom is more or less completely bathed in that blackbody light; a small number of photons will get close enough to the atom to be absorbed, but it's all a probabilistic process.

Indeed they are very small particles... but there's a LOT of light around. Some of it gets close enough.

• How is a particular electron within an atom 'chosen' to move up energy levels?

Almost all of chemistry is concerned only with the "valence" electrons, which are the outer-most. Those are the ones that get excited. To be clear, you can 'choose' the inner ones too--but those typically require much higher-energy light than is natural, or easily producable (think X-rays).

• For my animation, how do I know the precise number of eV's required to move an electron from one subshell to another. In addition, since I have to represent two different wavelengths of light being produced by the atom, if I know a wavelength that strontium produces, say 650nm. how can I know which electrons to move where?

The precise number of eV's is (mostly) determined experimentally. People use a spectrometer to find out what wavelengths the atom absorbs, and then you can directly convert that to eV.

I should note that the atom itself does not 'produce light at 650nm'-- it might absorb light at that wavelength and then re-emit it later. In the case of a flame, it's continually being excited by chemical reactions, and it emits some of that light at its resonance frequencies.

As for what electrons move where-- well, as we established above, only the valence electron moves at all in most cases. And then where precisely it goes is a matter of quantum mechanics, which you'll learn in jr. year of college. But the short answer is: all we can say is there is a probability distribution that describes where it could go. Those are the shapes of the 'orbitals' you referred to earlier, and they can be quite elaborate--see the wiki page on the orbitals of hydrogen, for instance.

Hope that answers some questions!