The operating frequency makes a big difference, so it's definitely worth playing with. You could add a feedback coil to turn the driver transistor(s) on and off which will make it run closer to resonance.
I've never found a cheap source of usable fine wire either.
The sound is simply due to the coils (and core if not fixed) vibrating with the magnetic field, and the hiss is caused by the corona discharge.
I'm not sure connecting several together will work out. I would be interested in the results...
Be very careful not to breath the ozone gas that is created by the discharge - it is extremely poisonous. I've almost passed out from that before now, in less than a minute of having my head over a big arc. The odour of it is obvious.
To be honest, I always found using the salvaged secondary HV coils worked out better than my own wound ones.
Have (safe) fun! :-D
Edit to add: I use petroleum jelly to make quick firelighters (I do a lot of outdoors backpacking type stuff). I wonder what will happen to your coil if it gets a bit warm and the discharge hits some PJ vapour? :shock: