Tesla Coil audio music player circuit analysis
Simple induction.  I am amazed  and impressed that you have achieved this with low-voltage components.  But they is also why the CFL needs to e close to the transmitting (coupling) coil.

The Tesla concept is pulsed electric voltages fed into a coil that radiates it.  The switching determines the frequency of the pulses.  Traditional tesla coils use a spark gap so the pulses are very quickly turned on and off.  Your transister is doing that part for you.  The audio part of the circuit is frequency modulating the pulse stream so you can hear it.

Then the mercury atoms in the tube vibrate with the pulses, giving off UV energy.  The phosphor lining is what converts the UV to visible, and can be many different colors.  

Word of warning:  If you break a CFL be sure not to step on it barefoot.  Mercury is toxic.  

You asked about various other components, and since you say it is working I don't have time to completely analyze the circuit,  But that's the basic principle for how it works.

You asked about finding a middleman, and I would say you have found it.  I have done music feeds into Tesla coils but always used the sparks as the audio transducer so I could hear it.  That means using a smaller induction coil with very high voltages inducing currents into a resonant secondary coil.  

The difficulty with inducing music in those is that  you need a solid-state "spark gap" that can be frequency modulated at very high current levels.    That's a whole lot more work (and a lot more dangerous) than what you are doing.

Using near-field radiation into a CFL is a neat idea.  I'll have to try this myself.