Is there a rule of thumb on vinegar's abilities as a preservative?
is it solely by virtue of its acidic
  content?


Yes, not a lot of biochemistry is happy living at pH 2 - and the stuff that can live there is already happily living in your gut with a ph of 1


  is there a kind of ratio (or other metric) of vinegar to know whether something needs to be refrigerated to keep from spoiling?     


You mean foods already packaged in vinegar? It would depend on how well they had been cooked/sterilized before packaging and how strong the vinegar is.
Industrially made pickles will be a lot cleaner than something you pickle at home, but store bought pickles are likely to use much cheaper, more dilute vinegar than your grandmother did.


  what components can act to counteract vinegar's ability to preserve food?


Anything too alkali will neutralise the vinegar's acidity


  can vinegar stabilize anything that otherwise would spoil? 


Not necessarily, it has to be something where the vinegar can penetrate into the food - it's not going to preserve steaks as well as vegetables. And of course the vinegar taste is going to help if you are trying to preserve say strawberries!
Also fungal spores can probably survive dormant in vinegar (if not grow) - fungal spores can survive dormant pretty much anything.