In the circuit provided by the link, would having an open secondary coil (not having it connected to any ground) damage the it? The coil is connected to pulsed 240v ac at 50hz. If it is not arcing, (meaning it has an open secondary coil) where will the electric field collapse, and will it damage anything? Furthermore, if it can damage anything in the circuit, will it do so if it is under powered with a very low current?
The larger range of capacitors have lower impedance, and can conduct more primary current but secondary arcing, if sustained can melt the secondary fine wire but this can be protected by using carbon ignition wire used by sparkplugs.
Since Copper has a PTC tempco., the weakest section of the secondary may rise in temp then resistance rises possible leading to thermal runaway then fusing open.
If you want to minimize damage risks, then using a carbon ignition wire for secondary used for spark plugs and always use plastic caps the larger size reduces resonant frequency and increases current.
The sunflower oil improves the insulation breakdown of air by ~ 5x due to impurities in oil and components but in transformer grade oil up to 25x and also distributes cooling to hotspots.
This kit was not "engineered" with coil inductance and resistance but appears to work. As I recall , I could put 1V from a 50 Ohm signal generator and get 1kV out sine up to about 20kHz with no load, meaning a coil that looks like this one can have a turns ratio of 1:1000.
The magnet wire has various insulation breakdown ratings and impossible to tell what your model is rated for but ought to handle 30kV easily. However, I doubt oil immersion affects that rating being sealed.
The oil is safer due to protect from creapage surface breakdown as you are applying about 300Vp 100Hz pulses to the primary coil intended for 12V pulses. So always start from the dimmest setting.
A 1A fuse and foot safety switch might be suggested so if an accident occurs, you fall over and turn off the power.
Your ignition transformer appears to be a voltage transformer, and if this is the case, an open secondary will do no damage, however, be aware that current transformers exist as well, and must never be allowed to have an open secondary while operating.
A voltage transformer maintains a relationship between input and output voltage, so any current that flows on the secondary will be determined by the impedance of what is connected.
A current transformer maintains a relationship between input and output current, and when presented with an impedance on the output, it will produce whatever voltage is necessary to maintain this relationship. This means if the secondary is open circuit (Resistance tends toward infinity), it will attempt to produce a voltage that tends toward infinity to maintain output current, causing arcing or dielectric breakdown and damaging components.