This is sadly normal. Your loudspeakers have bass reflex loading, so at very low frequencies there's little restoring force on the cone, and any low frequency signals will move the cones.
The problem is that many phono stages in amplifiers, or indeed separate stages have no rumble filtering, so low frequency rubbish gets through unattenuated. This can be made worse by any mismatch between the cartridge's compliance and the arm mass, which can reduce the resonant frequency to warp frequencies. Ideally, the arm/cartridge resonant frequency should be around 9-11Hz, above the 3-7Hz warp frequency and below the 20Hz or so lowest audio frequencies.
I would firstly check that the arm/cartridge resonant frequency is around the 10Hz mark, using any of the commonly available test records. At first glance, I don't see any great mismatch between your cartridge and arm, but that depends on which version of the Rega 2 your NAD is a clone of. If it has the arm with a removable headshell, that may have been changed to a heavier one, and that could make the problem worse.
Secondly, looking at the 8000C manual, I don't see any mention of a fixed or switchable rumble filter, and with the phono frequency response quoted as flat (-0.5dB) at 20Hz, I doubt there is one, as otherwise I'd expect 20Hz to be down a couple of dB.
As to damage to your loudspeakers, it shouldn't be doing any damage, but is clearly reducing the voltage output capability of the amplifier, so you won't get as much useful (i.e. audible) power from the amp before clipping.
The real solution to all this is to buy an external phono stage with a built-in rumble filter, or change the pre-amp to one that has one, or just to live with it. Loudspeakers with sealed bass won't exhibit the wild cone movements as there's a restoring force all the way down to very low frequencies, but the problem of wasted voltage excursion will still be there, just rather less visible.
S.
