… I can also see with the DMM that all pins 4 and 5s share continuity - should this be the case? And if so, then why does each of them have an independent wire, one red and one black?
I think you’re referring to the EL84 sockets, and that this is with at least one 6.3V valve in a socket?
If so, the continuity is via the valve heaters, ie the heating filament inside each valve.
My understanding is that the grounding for the heater should be pin 9, which according to the schematic goes to ground via a 100R resistor, but I don't get any resistance reading between it and the chassis?
I think you may have misunderstood almost everything above.
Pin 9 of an EL84 is its screen grid, it’s at high voltage. Connected to their HT supply node via 100R resistors R73,75,77,79.
The schematic shows the centre tap CT of the mains transformer’s heater winding to be connected to 0V / circuit common / chassis / (ground).
Lastly - as I said above I don't feel like at any point during the first round of taking readings I shorted anything. But say I did, could this be the cause of the fuses blowing? And could the original lower level issue and the current fuse blown/high voltages/no volume issue be a separate one?
All the above is possible.
It’s a fact of life that probe tips slip at the most inconvenient moment, and damage can easily be caused.
I suggest to use clip on probe adapters, especially on output valve socket terminals.
Alternatively, use a file to get a sharp point on the probe, then sleeve off all but the very end.