Let's get techie.
The original line emitted by an isolated atom is always the same, limited in width to the resolving power of your method of separating the light into its component wavelengths. I.e. what is the resolution of your monochromator? But for the quiescent atom, there will be no noticeable variation in the wavelength between two photons. The difference is based on a transition between two quantized states, and the difference between those states will be invariant (according to current theory.) However, the problem isn't about a single molecule or atom. It is about a cloud of atoms.
Now let's examine the choices:
Rotation - can lead to Doppler effects - so there would be variation in wavelengths APPARENTLY emitted. You would see them as that central-line wavelength plus or minus the an amount proportional to the speed of rotation. The distribution of that frequency would be fully described by some trigonometric and probability distribution equations. In short, the width of the "blur" factor depends on the speed of the cloud. An intensity graph would be Gaussian, with the kurtosis of the Gaussian distribution proportional to the rotational speed. If the cloud were both rotating AND moving, the skew would be proportional to the rate of overall cloud motion.
Even in a confined but non-rotating space, there is this to consider as well: You expect to see a wider range of particle velocities as gas density increases, because the derivation of the PV=nRT gas law is statistical in nature. It is assumed in that derivation that you have a statistically normal spread of directions and velocities for your gas molecules. Which leads to a Doppler effect. Saying "statistically normal" again implies "Gaussian."
The addition of heat (temperature) further scrambles the ranges of molecular speeds such that the distribution cannot be ignored. Again, you would expect to see variation in speed and direction of gas molecules as they collide with each other. This would lead to that variation in the wavelength due to pressure or thermal effects and a Doppler shift. The statistical derivation still applies, thus predicting a Gaussian intensity pattern around the central wavelength.
Therefore I would suggest "all of the above."
The infamous "assume a spherical xxxx" jokes are based on that gas law proof, usually having to do with a ludicrous choice of xxxx. For instance, the physicist is making a killing at the race track, betting the Trifecta on every race. Someone asks him, "How are you picking so many winners?" To which the answer is, "Assume a spherical horse..."