It would be difficult to adapt the Tesla coil system described in that unconventional (it is missing some page information, probably CSS) web page to be smaller. It is using the method I mentioned in post #5 with a 'ringing' driver stage to improve the drive waveform to the MOSFET. It is possible to see the images if you follow the links in the page source but I suspect the author has abandoned it.
The exact frequency in that example isn't important but whatever you choose, trying to make it with 'small dimensions' is very risky owing to the danger of flash over. The Tesla coil principle relies on the voltage between adjacent turns being relatively low but it results in a lengthy construction. If you try conventional multi-layer windings to make it smaller I think you are asking for trouble.
If you can live with lower frequency, in the 15KHz region the problem is MUCH easier because transformers are easily available, ones rated to >25KV at suitably high power are used in conventional CRT TVs and monitors. It is a long established technology with excellent reliability and circuits are abundant.
Brian.