Different coil turns
Why there is not 8 turns on each side?

Because 6 is all that is needed to switch the gate. It's not like a push-pull driver which would be symmetrical.

So if I need 400V, there will be the same ratio? 5:3?

Yes. If you want to change the output voltage then change the number of turns on the secondary. Reduce to 4/5 = 80% of the turns required for 500 V.
I would not expect good voltage accuracy or stability from such a simple circuit.