Nixie Clock Power Supply
1: The 555 oscillates by charging up a capacitor to an upper threshold voltage, then discharging it to a lower threshold voltage, then starting again; the speed of oscillation can be altered by varying those voltages so that the same R/C values take more or less time to reach them. The thresholds are normally 2/3 and 1/3 of the supply voltage, set by three resistors inside the chip, but pin 5 is connected to the upper threshold so it can be "tweaked" by an external signal to alter both the frequency and the duty cycle. Increasing the control voltage on pin 5 increases the charging time by more than it increases the discharge time, so the frequency is reduced but, crucially, the duty cycle is increased. The following diagram from TI's 555 datasheet illustrates the effect of control voltage changes on the charge/discharge curves and the output waveform:



While the capacitor is charging, the OUT pin of the 555 is high and Q7 is turned on to "charge" the inductor. While it's discharging, the inductor charges up C4 to increase the regulator output. The regulator output voltage is determined by how much "charge" is put into the inductor and how often this charge is transferred to the output; the more charge and the more often, the higher the output voltage.

2/3: It's hard to separate these questions; R24, R7, R23, Q8 and R25 are used to provide a feedback loop to control the output voltage. The first three resistors reduce the 180V regulator output to approximately 0.6V in order to operate Q8 in its linear region. As a lower control voltage means a lower output voltage, Q8 is there to provide an inverting function, so that if the output voltage is too high, it turns on more fully, reducing the control voltage at pin 5 and hence the duty cycle so that less charge gets put into C4 during each cycle, reducing the output voltage. Conversely, if the output voltage is too low, Q8 turns off slightly and increases the control voltage and duty cycle to increase the output voltage.

4: Because a circuit diagram isn't really the place to put functional specifications, although voltages are often used as labels for supply connections.