Q:

Significance of Proton Tunneling

It has been repeatedly emphasized that proton transfers "involving" three or four membered ring transition states are not favorable and should not be drawn as the primary mechanism for any reaction. 
However, with regard to a recent discussion involving $\ce{H3CSH}$ it was noted that the rate of proton tunneling in the anionic form $\ce{H3CS-}$ was likely extremely high. 
Carbon-Sulfur Bond Lengths; Resonance Effects (Or Lack Thereof)
So:
a) Is this a valid rule of thumb: avoid intramolecular proton transfers unless the transition state has 5 or 6 members in its ring?
b) What's more significant with regard to proton transfer - proton tunneling or physical proton transfer? Or is the answer an "it depends"? 

A:

a) Is this a valid rule of thumb: avoid intramolecular proton
  transfers unless the transition state has 5 or 6 members in its ring?

Yes

b) What's more significant with regard to proton transfer - proton
  tunneling or physical proton transfer? Or is the answer an "it
  depends"?

It depends, at temperatures around absolute zero, tunneling can become significant.  At ambient temperatures or above, physical proton transfer is much more common.  
As to the methyl mercaptan anionic example you cite, to the extent that the various carbon-sulfur double bond resonance structures contribute, I would expect a lowering of the barrier around carbon anion planarization - and consequently a lowering of the barrier to inversion at the carbon atom in the anion.  Therefore, it is not clear to me that tunneling of the sulfur bound proton would be preferred to carbon inversion in order to bring about the syn-anti interconversion, especially since any presumed experiments to generate the anion would not be performed near absolute zero.

