No, atoms do not vibrate, molecules vibrate through motion of the atoms about a common center of mass. This motion is determined by the connectivity of the atoms through chemical bonds, which can be thought of like "springs", each with an associated spring constant. Thus, it is usually a good approximation to treat the vibrational motion of molecules as harmonic, at least at low energies.
I have no idea what that sentence was intended to convey. The vibrational motion of a molecule in the harmonic limit can be expressed in a complete basis set of orthogonal normal modes. Each normal mode can be represented as an independent harmonic oscillator, and so in this limit, it is completely reasonable for all of the modes of the molecule to be excited to any arbitrary vibrational state (i.e. quantum number). For example, in the harmonic limit, for the OP's 3-mode molecule example, you could easily have 1 quantum in each mode, or two quanta in each mode, or 3 quanta in one mode and zero in the others. *In the harmonic limit*, those states would be stable for an indefinite period of time (i.e. they are eigenstates). Of course, in any real molecule, the modes are not completely harmonic, which leads to anharmonic coupling terms that allow energy to flow between the modes over time, this is the well-known phenomenon of intramolecular vibrational redistribution, or IVR.
That is not completely true, the normal modes *can* be excited by electromagnetic radiation (i.e. photons), however they can also be excited (or de-excited) by collisions with other molecules. This is called T-V energy transfer, (translation to vibration).
No, the vibrational spectrum of molecules ranges from about 50-100 cm-1 (the terahertz regime), for large amplitude motions of heavy molecules, up to about 4000 cm-1, for stretching vibrations of light, strongly bonded molecules like H2 and HF.
Ok, I agree with that completely
No, this is not true. Vibration is *completely* dependent on temperature. You can define a Boltzman population for each normal mode that describes the probability that a given quantum state will be populated at a given temperature. These factors increase with temperature, and thus so does the average internal energy for an ensemble of molecules at a given temperature. In fact, all molecules have a thermal dissociation threshold, which is the temperature where the average internal energy of the molecules is large enough to break the weakest chemical bond in the molecule.
It is a semantic point ... it makes sense to say that vibrations arise from atomic motion, or that vibrations arise from motions of atoms in molecules (or materials). However, I don't like the term "vibrations of atoms", because that makes it sound like the atoms have an intrinsic *internal* vibration, which they don't. From your posts, it sounds like you have a background in solid state/materials, so in that case I think it could be correct to say that vibrations in materials arise from "vibrations of atoms around their lattice sites", or something like that, where it is clearer what you mean from context.
Nothing, I wasn't disagreeing with that part of your statement .. only the part about requiring a photon for the excitation. As I say, vibrations can also be excited via collisions of molecules with other molecules (or materials, i.e. surface impacts).
Fair enough .. I wasn't thinking about solid state vibrational spectroscopy. Since the OP was asking about 3-mode systems, I automatically thought of small molecules, not solid state samples.
That is only true in the limit of perfectly harmonic vibrations, which never actually occurs either in molecules or in the solid state. With real systems there is a slight anharmonicity, so the "energy" you are talking about (I would call it the fundamental frequency of the mode), decreases with increasing quantum number. Furthermore, there is anharmonic coupling between the modes as I described, and so energy that is initially deposited in a particular mode will "leak out" into other modes in the molecule, generally quite quickly, on the timescale of a few vibrational periods.