The speed of sound is a single value yet the speed of atoms is distributed over many values. Does the sound wave front smear out?

It does, but the effects are negligible in the regions we think about.

If you think about a volume of air as a box of atoms bouncing around, you can apply an oscillating pressure gradient across that box and show that it behaves close enough to an ideal wave propagation medium that you can get away with using such an ideal model. The variations you are looking at "smooth out" on a timescale much shorter than the timescale of the sound wave being transmitted. This is a case where the central limit theorem is quite useful - you can basically show that the variance of the statistical medium you are thinking of is sufficiently negligible when occurring over the timescales we think of when we think of sound waves. That's not to say the effects you are thinking about don't occur, just that they are small enough compared to other effects that we can get away with handwaving them away and still have a useful predictive model left over.

The term used for this is "relaxation." The assumption is that the stochastic system you bring up "relaxes" fast enough compared to the behaviors we care about that we don't have to concern ourselves with those details. The random behaviors obscure any information that might have been held in the exact structure of the medium. All that is left is a homogeneous system which, because of the central limit theorem and large number of particles, behaves almost as an ideal wave propagation medium.

This assumption is not always valid. There are times where you need to use a more complete model, which includes the statistical model of the air molecules. One particular case where we have to do this is when dealing with objects that approach the speed of sound. As you approach the speed of sound, the assumption that the stochastic effects are on a short enough timescale that we can ignore them starts to fall apart. The timescale of the events we care about start to get closer to the relaxation time of the stochastic system of particles. Now we have to account for the sorts of effects you are looking at, because they have a substantial effect. Now we start seeing behaviors like shock waves which never appeared at lower speeds.

We also have to start considering more complete models when dealing with very loud sounds. Once a sound gets above 196dB, you cannot use the nice simple ideal wave propagation formulas because the low-pressure side of the wave is so low that you get a 0atm vacuum. Modeling this correctly requires including effects that were not in the simple model we use every day for normal volume sounds at normal speeds.

They do, its just usually negligible in practice. There's also scattering because the particles are not all the same (H2O, N2, O2, etc.)--but that, too, is usually negligible. Its mainly because there are so many particles in a single wave. Consider that the wave must be extremely short before it becomes noticeable (megahertz).