In the 2D case, there's no escaping exponential decay of the autocorrelation function for any observable satisfying certain regularity properties. (I'm not sure if this is known to be true in higher dimensions. If it's not, then there could conceivably be traps with sub-exponential escape times or even attractors, but I'd be surprised if that's relevant here—I think it's just hard to prove.) Sticking to 2D, the question is just how the time constant in that exponent for the observable in question compares to 20 seconds.

The presence of persistent collective behavior is a decent intuition but I'm not sure it saves you. I'd start by noting that for any analysis of large-scale structure—like a spectral analysis where you're looking at superpositions of harmonic sound waves—the perturbation to a single particle's initial position is a perturbation to the initial condition for every component in the spectral basis, all of which perturbations will then see exponential growth.

In this case you can effectively decompose the system into "Lyapunov modes" each with their own exponent for the growth rate of perturbations, and, in fact, because the system is close to linear in the harmonic basis, the modes with the smallest exponents will look like the low-wave-vector harmonic modes. One of these, conveniently, looks like a "left-right density" mode. So the lifetime (or Q factor) of that first harmonic is somewhat relevant, but the actual left-right density difference still involves the sum of many harmonics (for example, with more nodes in the up-down dimension) that have larger exponents. These individually contribute less (given equipartition of initial energy, these modes spend relatively more of their energy in up-down motion and so affect left-right density less), but collectively it should be enough to scramble the left-right density observable in 20 seconds even with a long-lived first harmonic.

On the other hand, 1 mol in 1 m^3 is not very dense, which should tend to make modes longer-lived in general. So I'm not totally confident on this one without doing any calculations. Edit: Wait, no, I think it's the other way around. Speed of sound and viscosity are roughly constant with gas density and attenuation otherwise scales inversely with density. But I think it's still plausible that you have a 300 Hz mode with Q in the thousands.

I think you're probably right. It does seem plausible that there is some subtle structure which is preserved after 20 seconds, such that the resulting distribution over states is feasibly distinguishable from a random configuration, but I don't think we have any reason to think that this structure would be strongly correlated with which side of the box contains the majority of particles.

Predicting the ratio at t=20s is hopeless. The only sort of thing you can predict is the variance in the ratio over time, like the ratio as a function of time is $\mu \left(t\right)=0.5+ϵ$ , where $ϵ\sim N(0,{\sigma }^2)$ . Here the large number of atoms lets you predict ${\sigma }^2$ , but the exact number after 20 seconds is chaotic. To get an exact answer for how much initial perturbation still leads to a predictable state, you'd need to compute the lyapunov exponents of an interacting classical gas system, and I haven't been able to find a paper that does this within 2 min of searching. (Note that if the atoms are non-interacting the problem stops being chaotic, of course, since they're just bouncing around on the walls of the box)

Tangential. 

Is part of the motivation behind this question to think about the level of control that a super-intelligence could have on a complex system if it was only able to only influence a small part of that system?

There are a lot of assumptions in "omniscient forecaster knows the position and velocity of all molecules at t=0" that make the answer "probably possible to calculate, probably not in real-time on current hardware".  

Edit (motivated by downvote, though I'd have preferred a textual disagreement): I actually fight the premise.  "omnicient forecaster" is so far from current tech that it's impossible to guess what it could calulate.  Say it only has 32 bits of precision in 3 dimensions of position and velocity, so 24 bytes for each of 6x10^23 particles.  1.4x10^25 bytes.

Call it 2 yottabytes.  There's no way we can predict what such a being might or might not be able to calculate, to what precision.