Classical, Rock, Heavy Metal, Punk, Country and Western, Electronic, Grunge, Pop, Folk, Opera (is this classical?), Fusion Jazz, Jazz, Chamber (is this Classical?), Blues, New Wave .... what have I left out? What have I forgotten?

Well, well, pretty good classification already, so yes, maybe the main genres are classical, blues, jazz, electronic, new age, hip-hop, country, folk, rock, pop, reggae, soul, metal, heavy metal, alternative .... and way more like punk, progressive rock, alternative rock, indie rock, hard rock, ( ok, ok, and a bunch of other sub-genres in rock) goth, grunge, anime... We can see many many sub-genres in all of the genres, like in blues there are least, don't know... 20 or more sub-genres, as well as in jazz, or metal, pop... in Latin music, pffff, so many sub-genres also. Also, we have music categories by cultures, nations, continents (Asia, so many different kinds of music... Africa too)
In Europe, and all parts of the world that were colonies once, we talk about occidental music and all these we mentioned here are actually part of it and came out from occidental culture and music. The word occidental can be confusing nowadays, but yes, all classical music is part of it, starting with medieval, renaissance, baroque, classical period, romanticism, and then contemporary, from where everything else was developed!

Answering your questions, are opera and chamber music classical. Yes, opera is part of classical and chamber music too. In classical the two main branches are instrumental music and vocal (and the mix of them - vocal-instrumental music) - so, opera is a form of a composition expressed through vocal-instrumental music... I hope in English this word makes sense, I am translating in my head literally hahaha. I have studied instrumental music interpretation, sub-category keyboard instruments, specialty piano.

Oh, electronic music and all the branches of it... let us leave all those classifications to @edje :D

What a messy comment I made here 😂