That's normally done by electronics in the device switching off before the battery is completely flat. If you run a lithium battery completely flat it will never work again.
That's correct.
It doesn't prohibit anything.
Back in the days when MK were a reputable manufacturer, they used to offer a plug that would take 2.5mm² flex - I well remember them, as a lecturer at the college where I was a technician had me make a number of 2.5mm² extension leads with them.
There is no point in inspecting the innards of the toilet seat. The fault is almost certainly in the wiring at the socket or between the socket and the FCU.
Bernard is correct.
The wattage will make no difference.
When you fitted the latest bulbs, were you careful to not touch the envelope with your fingers? Fingerprints on the capsule can cause it to shatter as it heats up.
Depends on the SMPS BAS. Some just chop the incoming supply and rectify the chopped waveform. Even the others will have a spiky current waveform, since current will only flow when the incoming voltage is greater than that in the smoothing capacitor.
Have you read the reviews? One says "Both are faulty in that even with the sensor covered in 3 layers of tape and a thick wooden glove over the sensor and sensitivity set to the lowest setting the lights are activated." Presumably he means a woolen glove, but even so...
Are the connections OK? A cable fault could be something cutting into it, such as a sharp edge on a backbox, a nail or screw into the cable, rodent damage, etc, or even a badly manufactured length of cable - Atlas Kablo comes to mind.