A diode buffer like this will work, just supply th 555 with the output of this ....
https://wiki.analog.com/_media/university/courses/electronics/text/chptr7-f1.png?w=600&tok=d97e78
As far as driving a MOSFET with a 555, two mosfets may not be the way to go since there will be terrible current shoot through possibility. The way you have it might be the best 555 solution since you are using a push pull driver approach.... the key though is going to be driving T1 at the resonant frequency of T1 and C4. 15kHz seems too high for a transformer normally rated for 50/60Hz.... maybe 5kHz but that could be pushing it. You need to determine the inductance of the transformer and know that it will change based on the loading on the output.
As far as the micro, the key to preventing shoot through is to allow one transistor to turn completely OFF before you turn the other one ON. You can do this by either monitoring the voltage or waiting some arbitrary amount of time that allows the transistor to stabilize.
Making your own HV coil can be difficult. Do a search for how other HV transformers are constructed. Typically there are multiple coil stages... each stage may have 100 to 500 turns on it, but there could be as many as 10 stages ... the idea is that each stage can only generate a certain amount of voltage, but because each stage is separated, the voltage isn't enough to break down from stage to stage. With high voltage potential on the outer most stages, you must consider possible alternative high voltage paths, like the core for example... the core becomes fair game as a conductor and precautions must be taken there as well to prevent arcing. Not that you can't DIY and roll your own HV transformer, but HV behaves in ways you might not expect if you are not familiar with HV. In a nutshell, anything that does not conduct electricity essentially becomes a storage device.
There may be OTHER easier ways to generate the voltage you are looking for....
Note: That multiplier looks like a Cockcroft Walton multiplier .... (see video links below) I'll see if I can find the build instructions for the capacitors. I used tranparancy film and adhesive aluminum tape
7kV - Cockcroft Walton multiplier powered from a 12V supply
21kV - Cockcroft Walton multiplier powered from a 12V supply