Electronic – Tesla coil, where do the sparks from the secondary coil ground to

It should be fairly obvious that you need to lay 1200 wires side-by-side in a space of 13.5 cm, so each wire (including its insulation) needs to be no wider than

$$\frac{13.5 \text{ cm}}{1200} = 0.1125 \text{ mm}$$

This would correspond to roughly AWG 38 with about 6 microns of enamel coating. I have no idea what the actual thickness of magnet wire coatings are, so you may need to adjust your figures to suit. Some online sources suggest that it's on the order of 0.0002 in., which would be about 5 microns.

The Slayer oscillator is actually a crude form of a resonant power supply. It runs at a resonant frequency. The resonant frequency is determined by L and C. Switching losses are reduced so that a BJT will run cool at sensible frequencies. You are stepping up so this makes L big and lots of turns make C significant. R has nothing to do with frequency. R is just for starting.

The transformer does have parasitic capacitance to ground. That is why C is drawn the way it is. If there was no parasitic capacitance to ground the oscillator would not work because its feedback path would be broken. Your diode does get reverse biased because AC current is flowing. I have used a red LED in place of D1 to indicate operation and give higher negative voltage on the transistor base to improve turnoff and possibly switching losses.

On your circuit, making R too small will actually make starting unreliable. You can ballpark R to give a small current that gives reasonable beta for Q. If your step-up ratio is too high, the oscillator won't start because the total gain would be less than unity. Make the step-up ratio N 10 or less so the oscillator won't need an exotic or nonexistent transistor.