Voltage Multiplier Frequency and Waveform
Each stage of such a voltage multiplier stacks a voltage the size of the input voltage on top of the circuit input voltage. But also, each stage has substantial losses. So the key to success is starting with an input voltage as high as possible.
Don't have it 1V but at least 200..300V. You can create that input voltage for the voltage doubler ladder using a flyback converter, which also has the nice feature of already having a &quot;square&quot; wave on output. The diodes and capacitors  each have to withstand the double input voltage.
The diodes have to be fast recovery types. 50Hz diodes as the 1N4007 won't work with high frequencies. This also limits the frequency of your input.

Important for your life (and that of others)
The input square wave gains its &quot;relative safety&quot; even at 300V from its high frequency —your muscles cannot follow, so you won't stick to it– but this doesn't apply to the voltage doubler ladder. Each of the capacitors is charged to 300V, which can reasonably harm you.
So choose those caps as small as possible. 50nF is a good value for small sparks.
Parallel each individual cap with a 10MΩ resistor to discharge it as soon the input voltage is off. Discharge may take a few seconds.
Same for the output. It can substantially harm you. Use a HV resistor of at least 10MΩ, if you can't get your hands on one, use several 2.2MΩ resistors in series.

To address your questions

As soon there is a spark, output voltage will drop quickly. That resistor limits the output current and that way, increases the time there is a spark. This is actually useful for studying it.
The charge stored in the capacitors depends heavily on their ESR and ESL values, the AC parameters of the diodes, and the frequency of the input. Their capacitance only puts an upper limit on the stored charge.
No practical ones. The waveform of the input current will be spiky in any case, because the capacitors can only be charged when the outside voltage is higher than their own voltage. As with any rectified power supply.