"How would you go about proving that something exists if it's not a physically observable phenomenon?"

You can't. Does a non-physical, non-observable phenomenon exist? Sounds awfully like 'nothing' to me! As for thoughts - well they are communicable (we're communicating them now) and I'm sure that some thoughts have measurable, physiological effects on the human body (let's not delve too deeply into that!).

"And if it's not separate from anything else, but intrinsically involved in everything?"

Well atoms are not separate from everything else and involved in everything. I think I'll pass on worshipping atoms, thanks!