'No news is good news'

So shouldn't it be "The lack of exceptions proves the rule"?

Or is it an archaic sense of "prove" that actually means "test" or something.

yes.

Not really (and I only found this out a couple of weeks ago). It's more like and what says. It's kind of a circular logic. For something to be an exception, it means there must be something there to be an exception to. If you see an exception such as "Only guide dogs allowed on the premises", that's an exception that proves the rule that, in general, dogs are not allowed.

(but yes, it is usually just used in a folksy way that doesn't really mean anything, so no wonder everyone gets confused)

It's difficult to use in a non-folksy way...

Otherwise professional philosphers would use it a kind of joker card : "My Grand Unifying Theory of Everything is clearly true - your objection being the exception that proves the rule."

Yeah, in philosophy, it would be uselessly circular, but not when there's any kind of authority making the rules (as with the guide dog example).