The absorption of IR radiation is due to vibrations of molecules. When a vibration causes change in charge distribution (or dipole moment to be more specific) the IR radiation is absorbed.
Generally, hetero-polar molecules, like $\rm H_2O$ and $\rm CO_2$, have permanent dipole moment. The external oscillating electric field in this case perturbs the Hamiltonian and causes IR absorption. Hence, they contribute to "Green House effect" by absorption of heat.
$\rm H_2O$, which is non-linear molecule, has three fundamental modes of vibrations. Symmetrical Stretching, asymmetrical Stretching and scissoring (bending). $\rm CO_2$, which is linear molecule, has four fundamental modes of vibrations. Symmetrical stretching, asymmetrical stretching and two degenerate scissoring modes, in planes perpendicular. The symmetric stretching mode in $\rm CO_2$ does not produce or absorb any IR, as it does not cause change in dipole moment, but other modes do change charge distribution causing absorption of IR.
Well, the homo-polar molecules, like $\rm N_2$ and $\rm O_2$, does not have any permanent dipole moment. The external oscillating electric field does not perturb the Hamiltonian for nuclear motion, and does not absorb IR, though it perturbs Hamiltonian for electronic motion. Hence, do not contribute for "Green House effect".
At any point above or below the Earth's surface the atmospheric pressure is equal to the weight of air above a square metre surface. So if you go down a mine the atmospheric pressure goes up because there is more air above you.
If the density of the air varies with height as some function of height as $\rho(r)$ then the pressure at a height $h_0$ will be given by:
$$ P(h_0) = \int_{h_0}^\infty \rho(h)g(h)dh $$
where $g(h)$ is the gravitational acceleration. Although this looks simple enough the density of the air varies in a complicated way because it's affected by temperature differences. As you down e.g. a mine the temperature goes up and the air density goes down. I found some calculations of the pressure change in this paper, though the authors bemoan the fact that no actual experimental measurements exist in the literature.
Re your other questions, I believe the upper pressure limit is due to the toxicity of nitrogen at high pressures rather than your ability to breath. The proportion of oxygen in the air wouldn't change (much) and since the density is increasing the amount of oxygen per cubic metre of air would indeed increase with depth. I doubt pressure changes would be detectable on your skin, though most of us have detected pressure changes when our ears pop.
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From your comments it seems that effectively you are asking about "why do gases mix so easily?"
If a system such as a mixture of gases is kept under constant temperature in a constant volume, the equilibrium state corresponds to the minimum of Helmholtz free energy:
$$A = U - TS$$
As you see, for $A$ to reach the minimum either the energy $U$ should decrease or the entropy $S$ should increase (or both in reality).
Minimizing energy. Most of the energy of common gases at normal conditions comes from their kinetic energy defined by the temperature. Energy due to intermolecular potential is negligible. So the only possibility to lower the energy is to lower the gravitational energy. In essence it would require the mixture to perfectly separate --- heavy gases at the bottom, light gases up.
Maximizing entropy Maximum entropy for the system in hand (under specified conditions) would imply perfect mixture, the state of most disorder. That's actually what drives the diffusion.
So as you see, the equilibrium state is a compromise between low energy and high entropy. For gases the entropy wins, because there isn't much energy difference between a mixture and a separated state (apart from gravity, which is still small).
As for your example with oil and water the situation is opposite. Unlike gases considerable amount of energy in liquids comes from intermolecular forces. Thus there is a huge differences in energy of interaction water-water or water-oil, so it is more preferable do separate to considerably minimize the energy.