Very scary circuit:
* No power transformer means the chassis is connected directly to the power line, relying on correct house wiring to keep it at the neutral instead of the hot voltage, and to keep the neutral at least close to earth ground. I would not even take it out of the console without an isolation transformer in place, and a grounded chassis.
* A single rectifier means it draws a net DC current from the power line, and thus will unbalance the power transformer of anything else connected to that half of the power panel. It also feeds rectifier reverse recovery spikes directly into the power line where they can be piked up by any other audio equipment nearby.
Now, questions:
>> 1.) Why does it seem to need some treble roll-off between the driver and power stages? ...
Hard to say; there seem to be many equalizations in the circuit, which may compensate the source, transformer, speaker, etc. Not knowing the characteristics of the output transformer or speakers, I'd want to measure the frequency response, possibly even at intermediate points, to make a guess as to why each one is there.
>> 2.) Are C57 and C62 (the caps across the output transformer primaries) there to provide negative feedback? When I remove them entirely, there is audible distortion in the high frequencies. As I increase the value, I get relatively more bass response and less treble (does not sound good), and when I decrease the value, the amp sounds more open and lively than with the stock value, but distortion creeps into the treble.
It is not feedback. It may be to prevent transformer leakage inductance and/or speaker impedance rise from rendering the plate circuit unstable. Beyond that, see above.
>> 3.) Can anyone explain the output transformer primary tap/power supply arrangement to me? I have gathered from internet research that this is some kind of hum-cancelling arrangement, but I don't understand it.
Yes, a portion of the output transformer serves as a choke in the power supply, where the hum that it reduces in that application cancels some of the residual hum at the power supply output. It also cancels some of the DC in the primary winding, allowing a smaller and cheaper output transformer. Since you have changed the power supply capacitor values, the hum balance no longer works as intended.
>> 4.) The 7695's that were in it to start with are fairly quiet, but when I put any other 7695 it's noticeably noisier. Is this just tube variability among manufacturers/tubes, or is there something about the circuit I could adjust to improve it?
Since the only change is different tubes, it does see likely that it's due to tube differences. Without knowing what the differences are, or what the noise is like (hum/buzz/hiss/rushing/rumbling) it's difficult to guess what might be happening.