Does expansion exist on the micro level as well as the galactic level?

[Ich]

Now what happens if this chain goes on beyond the Hubble radius? This would mean the last star would be moving faster than light with respect to stars in its neighborhood that are stationary with respect to comoving coordinates.

I think I owe you an answer, too.
In your thought experiment, better let the stars be held in place by some SF drive system. If they were at rest because of gravity, there would never be a necessary end to the chain.

Starting from a free falling star, the chain can go on until it reaches an event horizon. In a vacuum dominatet universe, that would be at the Hubble distance. In all other universes, the horizon is further away or does not exist at all. The Hubble distance has no meaning then.

Example: in an almost empty universe, a star at Hubble distance would need a peculiar velocity of 0.76 c, not 1 c. The definition of cosmological coordinates is such that the so-called recession velocity is in fact a recession http://en.wikipedia.org/wiki/Rapidity" .

It would be good to clear up the definition of proper distance since I thought it meant as you did, simply the coordinate system that shows no redshift.

It isn't. proper distance is an explicitly time dependent concept, these are not static coordinates.

Could someone clear up whether understanding the actual nature of expansion is related more to SR, in that distances and time change with the relativistic speeds of the moving distant galaxies, or does it have to do more with GR in that the spacetime over large distances is stretching, or both, or neither?

Both. At larger scales, gravity becomes important, so you need GR for an accurate model.
Then, in SR, you're not free to choose your coordinates as you like, you have to stick to the standard ones. In GR, you can use cosmological coordinates, which express the symmetries of spacetime much better.
I think what you mean is not SR/GR, but rather whether you express things in almost static coordinates or use dynamic ones. Both is GR, both is accurate. Static coordinates are close to what we're used to, you have gravitation there and velocity and such, which is very helpful. Dynamic coordinates are, in complicated cosmologies, much easier to calculate, and reflect better the symmetries of spactime. They are prone to misconceptios, however. Not in themselves, but because people - including quite a few cosmologists - tend to think of them as being close to what they're used to. Forget that. If something is supposed to be stationary wrt something in dynamic coordinates, it's a safe bet to assume it's moving. That's obvious for comoving distance, but it's true for proper distance also.