Does the molecules in a gas approach a common speed over time? 
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*and if they do - how does that actually work, if we look at the individual molecules 'bouncing around and about'?
To my untrained eye, it appears it would require speedier molecules to collide with multiple others in a single instant (is that even possible?), or perhaps transforming some speed into rotation?

 A: What in fact happens is rather the opposite of what you suggest.
There is good experimental corroboration that a gas, when initially not in the state of highest entropy, will relax to Maxwell-Boltzmann distribution
At the heart of the concept of Maxwell-Boltzmann distribution is that the state of the gas is characterized by the molecules moving with a range of velocities. The further away from the average the smaller the probability of a particuler gas molecule to have that outlier velocity.

I surmise from your question that the concept of relaxing to a distribution of velocities is counter-intuitive to you.
The natural question is of course: how can an individual gas molecule acquire a velocity that is significantly higher than the average?
Here is a way to understand that:
At every point in time there is a certain probability for a portion of the collisions that occur to be a collision close to a 90 degrees angle. Let's say molecule A is already at an above average velocity, and then it is impacted sideways by molecule B. The sideways impact means that molecule A does not lose any of its existing velocity. The resulting velocity of molecule A after that collision event will be the vector sum of the pre-collision velocity and the velocity component imparted by the collision. That is: after a sideways collision the velocity of molecule A will be even larger than pre-collision.
This show how it is possible for a sub-population of the gas to have a veloicity that is higher than the average.
Still, for any above-average velocity molecule the probability of losing that surplus velocity again is larger than 50%. There will be a constant churn.
For the Maxwell-Boltzmann distribution:
The state of highest entropy is the state of highest probability. The state of highest probatibility is one with a distribution of velocities.
