Next thing I'd try is to remove the speaker from the cabinet again. Examine the tinsel wires between the woofer terminals and the cone, and see if they touch when the woofer is pressed in. Check inside the cabinet, make sure there's no way that the stuffing could be causing those wires (or anything else inside the cabinet) to touch the cone. Then try hooking up the speaker wires directly to the woofer, lay it on its back, and run some music through it. See if it still makes those sounds. If it doesn't, there's probably something in the cabinet that's causing it to happen; if it does, then it's the woofer.
One other thing you might want to do is play a slow, low frequency sweep through the woofer so you can narrow down the frequency at which it happens (it may be also be a range). This will make it easier to reproduce. The fact that it doesn't happen at 30 hz leads me to believe that it's probably not a voice coil centering issue.