Electronic – Confused about the way this speaker is wired

Depends which way round it is. It's almost completely out of the main magnetic circuit, but its fringes will either add, or subtract, a little - maybe 10-15% - to the flux in the field where the voice coil operates.

If it subtracts from the flux the speaker will be slightly - paradoxically - louder (the same way an electric motor goes faster when you reduce the field strength - more motion is required to generate the same back EMF). Probably about 1dB louder which is quite difficult to hear. (Some preamps have digitally generated 1dB steps so you can test what 1dB sounds like, and yes it is just audible).

If it adds to the flux, the speaker will be quieter - but its bass will be better controlled.

I have seen some poor guy flown from Japan to Europe to spend a few weeks supergluing magnets onto the back of 5000 drive units, to increase the flux where the original magnets were weaker than the manufacturer had promised.

As pointed out in other response, it is heat that usually causes speaker failure. I recommend this article for some additional details on physical failure mechanisms: https://sound-au.com/articles/speaker-failure.html

There are a few other details worth mentioning, since your original question asked about what frequencies are more likely to cause failure.

Low frequencies (relative to the driver in question) can often cause failure because, for a given sound power output, lower frequencies require greater cone excursion. If excursion becomes so great that the voice coil actually leaves the magnetic gap, a magnetic short is formed and lots of heat is produced. Drivers usually have a specified maximum excursion, called Xmax. Speakers designed to produce lower frequencies generally have much larger Xmax values so they can deliver sound power at low frequencies. At any rate, care must be taken to insure that the driver is not driven in such a way that the cone excursion exceeds Xmax. This can be achieved by limiting the total power delivered to the driver and/or by applying a high-pass filter to prevent low frequencies with higher associated excursions from being delivered to the driver.

For different reasons, high frequencies (relative to the driver in question) can also cause failure through a mechanism known as breakup. Basically, standing waves can develop on the cone itself. Because cones are not designed with the resulting stress in mind, this can cause the cone itself fracture.