Does the capacitor play any role in the setting of the voltage at the cathode?
Not in setting the voltage but in keeping it constant when a signal is applied to the grid. At idle the voltage will be the same cap or no cap.
When you apply a signal to the grid, you allow more or less current to flow. As you draw more or less current through the 250 resistor according to V=IR the cathode voltage will go up or down. The cap takes the signal ripple out of the voltage and keeps the cathode voltage constant.
Think of a preamp tube, preamp tubes are cathdoe biased.
As more current is drawn through the cathode resistor the voltage on the cathode increases and the grid to cathode voltage changes. With no cap as the grid voltage increases, more current goes through the cathode resistor so the cathode voltage increases too. Now the difference between grid and cathode is not 1V even though you're going from -2V to -1V on the grid. Because the voltage is changing on the grid
and the cathode that 1V change does not appear to the tube to be a 1V change. Negative feedback. Going from -2V to -1V on the grid, with a cap keeping the cathode voltage constant, the change in the relationship between the grid and cathode is 1V: more gain.
With a smaller cap the higher frequencies are grounded, or to be more correct the cap can't charge and discharge quick enough to keep the voltage constant at lower frequencies so, the treb is accented.
But, does using a small cap boost treble? You can think of using a small cap as canceling negative feedback
on high frequencies, boosting treble, or think of a small cap as introducing negative feedback on lower frequencies lowering gain on low frequencies.
It's a little different with a PP output because as one tube draws more the other draws less.