Yeah, you can gently push around the dustcap and feel for rubbing in the voice coil, or you can hook it up to a frequency sweeper and listen for signs of distress...that's what my local speaker tech does. The battery test is usually to test to make sure you have it hooked up properly and in phase with other speakers, so I doubt there's much it can tell you.
BTW, you can always have the original reconed if it's broken.
And yes, it's possible that cranking the low end and boosting it with an EQ could blow it, but it would still have to be out of the range of its capabilities, and if you don't play too loud otherwise, seems a little unlikely.