I want to make some ballasts for lamps that aren't common where I am ie, 100W MV, 175W MV, 200W HPS, 310W HPS, 18W LPS, 26W LPS, 35W LPS, 55W LPS, 90W LPS and 135W LPS. I have seen an existing 18W SOX ballast but i can't wrap my head around how it works, I know that choke coils work by its impedance but the 18W ballast I found needs a capacitor to help strike the lamp and appears to be of the autotransformer variant does anyone know how those ballasts work? I have attached the image of the schematic of the ballast.

It is a choke with a winding connected between the lamp and mains, with a tap for the capacitor.
When the lamp is already lit, it works just as a choke, the capacitor does not play any significant role.
When the lamp is not ignited yet, you have a winding section between the capacitor and mains forming an inductor, in series with the capacitor. he circuit is off resonance, but still the impedances cause the capacitor voltage to be greater than the mains voltage, usually around 300..350V.
On top of that, the difference (Vcap-Vmains) is further transformed up by the ratio of the tap vs full winding, what makes the ballast OCV greater than the cap voltage (in the required 450..500V range).

The tap position is usually designed so, the capacitance required for the ignition voltage boost matches the capacitance then needed for the PF compensation when the lamp is running (the tap is closer to the mains side, so when running it is effectively connected across the mains).
The coupling between the winding is not made that much tight, as the leakage inductance helps to separate the lamp from the low impedance capacitor at the harmonics (generated by the lamp), so maintains the crest factor under control. But this property is not that critical, because the inherent ballast winding wire resistance is already quite effective in maintaining the current crest factor, it only means here is no special effort in making the coupling tight at all.

Could I use an autotransformer with an OCV of 300V and use the impedance of the autotransformer to limit the current to 0.35A and have the running voltage of 55V

It will work, but with higher losses than the trick with the capacitor.
The capacitor trick means the same losses as just a plain series inductor, so the total winding apparent power is 0.35A * 210V=73VA.
The autotransformer has primary winding winding that needs to have each winding rated the same, so the total is about 145VA. This apparent power is then bases for the power dissipation, as well as the size, mass and cost.

Today you may use an electronic ignitor to get the required OCV for the SOX start. These are able to generate the elevated voltage for startup yet keep the ballast as a clean inductor (so the lowest current crest factor, as well as lowest losses) during normal operation.

But all that was for a SOX ballasts, the other lamps require different style.