How to avoid from turning mushrooms brown while freezing it

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.

According to Alton Brown, talking about why a short-grain rice salad won't get hard when cold:

But it won't be hard, the way, say, refrigerated Chinese take-out restaurant rice would be. Why is that? Well, long-grain rice, which is usually what's inside one of these, contains a very, very high percentage of amylose. [holds up the model] Remember this guy? When this cools, the amylose and water come together to make a structure that's kind of like, well, kind of like a crystal, which is why it's hard as a rock. The process is called "retrogradation", and it's reversed when the rice is reheated. Now since medium-grain rices have more amylopectin in them, this never happens in the first place, which is why I use medium, and sometimes short-grain rices for all refrigerated-bound applications.

If you can't heat up the rice, have you tried making some sort of short-grain variety?