You can look at what I made for my CW multiplier. It consists of two AC flybacks in parallel driven by a ZVS driver. The ZVS driver was obtained through E-Bay and is very common. I also got the AC flybacks from E-Bay. They are less common.
https://highvoltageforum.net/index.php?topic=1524.0

Voltage and rep rate are related. The relationship depends on the available current from your transformer. You can get an idea by looking at the output of JAVATC. Specifically, look at the capacitor charge level output. This will show you the voltage level of the capacitor charge at the end of each rep just before discharge.

The most expensive component of any SGTC is the HV capacitor bank. There really is no cheap way around it unless you resort to beer bottle capacitors but those are very lossy and frequently fail.

For a small coil inside an apartment I would use a counterpoise. I would avoid connecting it to the house ground. The quality of that ground is unknown. You could end up destroying electronic equipment that is plugged in.

I thought that you just wanted the ability to change the output needle position according to some user-adjustable control. For that case, the "John Freau phase shifter" may work because you only need an adjustable phase shift with a 60 Hz frequency. However, if you want to see the needle rotate then you need some small frequency adjustment as others have stated. And as others have stated, some sort of inverter with 120 volt output is probably the most logical way to do it. I have seen "true sine wave" inverters for sale but they aren't cheap. I'm not sure if a conventional PWM (square wave) output inverter will work. You would just have to try one. In either case, you would then need a way to slightly adjust the output frequency of the inverter since they are designed to be fixed at 60 Hz. Depending on the method used to generate the 60 Hz output perhaps you could possibly hack into the circuitry to provide a way to adjust the output frequency slightly. I am not hopeful of that because I think that the output frequency is tied directly to the input frequency.

If the meter will operate properly with a PWM output inverter, you could use a single phase of the 3-phase output of a VFD. These commonly provide a way for the user to adjust the output frequency with a knob, either built-in or externally supplied. 240 volt single phase input VFDs with 240 volt 3-phase output are commonly available. I use one to control my lathe. I can adjust the output frequency with a knob. There may be a 120 volt VFD available but I don't know. If this would work I have seen small cheap Chinese VFDs on E-Bay for less than $100.

I just started researching this as you recommended. Is it possible to make one small enough for the lamp project? When I think of a Variac I think of a large Toroidal coil transformer with a large knob on it.

There are some very small variacs. I have one that is only about 3 inches in diameter. Consult E-Bay.

WAIT A MINUTE I think I just realized something that was confusing me in javaTC?

So for feeding a tesla coil DC like I'm going to do, the capacitance value of the resonant capacitor is only related to its tuning with the primary coil (making the primary + resonant capacitor frequency match the secondary resonant frequency)? In other words if I were to change the input voltage over a range of say 12-24V, it would have no effect on the required capacitance value of the resonant capacitor, I would just have to make sure it can handle the peak voltage?

That is correct.

The answers to that question are wide-ranging and would encompass an entire academic study which has never been done. In general, I would advise the following.

I wouldn't touch any Tesla coil streamer if it has an output power of more than a few watts. Those small Tesla coils which you see on E-Bay and plasma globes are OK. Even for larger coils, you may not feel much but the current is passing through your body. You don't have much sensation of pain because nerve cells do not respond very well to high frequency current. The current will also follow the paths of least resistance such as blood vessels and nerve fibers. Don't believe the skin effect myth that you may read about. The skin effect is only significant for excellent conductors such as copper. For a poor conductor such as the human body, the skin depth essentially extends through your entire body. The current flowing through your body heats internal organs, with the heating depending on the power level.

In my opinion, its just not worth risking your health just to demonstrate that streamers can strike your hand with no apparent effect. You are essentially experimenting with yourself. Would you feel comfortable with using your body as a wire? In effect, that is what you would be doing. Nerve damage can take years to manifest itself.

If indeed what you are seeking is information on transformer resonant frequency capacitor size, JAVATC provides an estimate. Its called "Resonant Cap Size" and is located in the "Power Calculations" output section. If you are using a NST, you don't want your primary tank circuit capacitor to be close to this value otherwise transformer damage can occur.

Bad tubes are always a possibility, especially if bought used. My first 833A was weak even though the filament glowed. I finally bought a NOS Ampex 833A and later a matched pair of NOS 833C tubes. Its always best to have two tubes on hand so that you can swap them out for testing. NOS 833A or 833C tubes aren't cheap though.