I can't think of a reason a shorted clutch switch would affect the audio but there again I've seen stranger things in my many years in the auto business, including a rear wiper that would turn on by itself every time I signaled a turn. After some head scratching and lots of wiring diagram tracing, I finally determined there was a low voltage back feed in the circuits that was tripping the relay that actuated the wiper. I swapped the relay and the problem was fixed. The original relay was defective.
If your clutch switch is shorted, your starter may very well turn over even if the trans is in gear with the clutch engaged. I would definitely recommend fixing that, and if you're lucky perhaps it will also fix the audio.
Jerry
Btw, there's a reason mom always said to not pull loose threads, strings, and other stuff. The other stuff in your photo is going to be a bear to shove back in.
