Some of these might work together but there's always going to be some exceptions.
I believe we already got some thousands of classical songs added as MB works.
Many soundtracks could have similar situation with classical thanks to soundtrack guideline (track artist=composer, recording artist=performer). Some film scores could be counted as classical but there's quite many jazz scores from 50s and 60s.
Not only limited to classical. Incidental music for theatre plays could be from any genre but would still have acts or scenes like operas. Danny Elfman composed music for circus performance (Cirque du Soleil) and this was divided to acts which were having names. Most of the Film/TV/Game soundtracks are having 2 levels if separate pieces are linked with master work, see for example video game soundtrack for The Great Giana Sisters.
It's common to use Italian tempo markings with classical but naturally same words can be used on any Italian titles. Usage in music isn't limited only to classical. Just try to search recordings having "Allegro" on their titles and you'll notice how bad this idea is. You could also try "Symphony" for useless results.
Hyperion might be some of the ones limiting only to classical but for example Deutsche Grammophon, Sony Classical and Naxos are also releasing some other stuff. It depends how wide our definition of classical is. Most of the biggest classical labels release film score recordings.
Wikipedia definition for classical music is "Classical music is art music produced or rooted in the traditions of Western music" so if we got some hundreds of years old African of Chinese music we might not want to count it as classical. Even in western music we got old folk music which commonly isn't counted as classical. How about old religious hymns? National anthems?
Jazz orchestras (big band) and studio orchestras performing for Movie/TV/Game soundtracks often have a conductor. Choirs also are commonly having conductors but not all choirs perform classical works.