Would an ideal gas be colder at higher altitude due to gravity? Since gas molecules are affected by gravity, wouldn't that make gas molecules at higher than average elevation slower (at the top of their ballistic parabola) and thus colder than air molecules accelerating to the ground?
 A: 
Since gas molecules are affected by gravity, wouldn't that make gas molecules at higher than average elevation slower (at the top of their ballistic parabola) and thus colder than air molecules accelerating to the ground?

In non-relativistic theory no, because in thermodynamic equilibrium temperature has to be the same everywhere. The slowing down does not occur on average because the molecules do not move along ballistic parabola, but collide with each other (in very rarified gas the collisions are rare and the establishment of the same temperature as down below may take long time).
In relativistic theory, yes because in thermodynamic equilibrium places with lower gravitational potential should have higher temperature (https://en.wikipedia.org/wiki/Ehrenfest%E2%80%93Tolman_effect); but the predicted difference is negligibly small for common gravitational fields like Earth's.
A: Alas, no.
Unless energy is trying to pass through the gas.
Then, yes.
If there is no energy input/output from the gas: "energy" may be transferred from top-down, but since a density gradient also forms, that energy is shared among more molecules. Net result is that the energy per molecule becomes roughly uniform and we have a uniform "temperature".
If energy is passing through the gas, under the influence of pressure/gravity, the gas forms convection cycles which quickly form a lapse rate (= -g K/km).
Note: The fact that lapse rate = g, tells us that convection is the main heat transfer mechanism in the atmosphere. Radiative transfer is a negligible component.
A.
A: Yes of course. In the atmosphere, pressure and density fall with altitude. So does temperature. It's all to do with the trade off between thermal and potential energy. Hence lapse rate $=\frac{\text{gravity}}{ \text{specific heat capacity}}$.
