1) I assume the connection from output to C56/R89 is providing some sort of (negative) feedback. The part left between the differential pair and PNP common emitter amp looks like a filter, is the purpose of the feedback to filter the signal?
Yes, those components form the negative feedback path that reduces distortion, sets the gain, and set the audio bandwidth. Without going into too much detail, the ratio of R89 to R92 sets the gain of the amplifier. C56 & R91 provide extra feedback at higher frequencies to reduce gain beyond the audio range which helps with stability. C55 rolls off the lower end of the audio spectrum and reduces gain to 1 at DC.
2) So, why is the cascode amplifier there in the first place?
The only reason for it, that I can see, is that it reduces the voltage across T12 & T13 (currently 90V each). If T12 was eliminated, and the collector of T13 was directly connected to the base of T15, then T13 would have 180V across it, which would double it's power dissipation.
3) The cascode amplifier output is inverted, so if it would be removed from the circuit, could the T15 base be driven with the same signal as T14 (ie. from the T16 collector)?
No... T15's base is at +89.3V and T14's base is at -89.3V so connecting them together would be disastrous. Technically, you are correct in that the audio signals are in phase, but DC wise it just doesn't work out for biasing very well. And since this is a DC coupled amplifier, using coupling capacitors to get around that (T14 vs T15 DC base voltages) would break the DC feedback path.
4) Are the common emitter amplifiers there to simply add more gain?
Yes. Gain and phase inversion.
5) Would it be correct to say that the push-pull amplifier is constructed in a Darlingtonish manner?
Yes. T9 and the T8-pair make a NPN Darlington. And the T6 and T7-pair make a PNP one.
6) Would the circuit still work, albeit with weaker output, if they were removed and the RX resistors connected straight to T7/T8 bases?
By "they" I assume you mean T6 & T9... Technically yes, though you'd have to change the bias circuit a bit (Short D39 & D21) and increase R75/R78... T15 and T14 will also get a bit hot trying to supply more current as you cranked up the volume. Magic smoke may leak.
7) I understand diodes D21 & D39 are there for the class-AB bias but what about NTC1, BIAS1 and the extra diodes? Are they just ensuring that T6 & T9 are active?
NTC1 is there to ensure that the bias current in the output stage stays steady as the amplifier heats up. The BIAS1 resistor reduces the voltage a bit to tweak the bias current a bit lower.
There are 4 diodes because there are essentially 4 base-emitter junctions in series that are being forward biased. (T9-T8-T7-T6)... Since a diode junction drops approximately the same voltage as a base-emitter junction, and that voltage tracks with temperature, they use 4 diodes in all.