Do I still need my iTunes directory? After upgrading to Catalina and the Music App I have two directories:
iTunes and Music

When I launch the app I noticed all my Music has the cloud icon (as I use iTunes Match) so its not downloaded locally. If I download a song it appears under Music/Media and not iTunes/iTunes Media.

I'm in the process of downloading all the songs in my library. Once I do this can I delete the iTunes folder and everything in it (including the iTunes Library file, iTunes Media, etc.)?
You are correct - both folders are not needed.

Once the new folder is set, you can consolidate all the media there, make one last backup of the old structure and delete it.

I’ve not seen them split often, so make sure you have a backup in case you find any old files didn’t migrate. Also, keep backing up - some people have felt that Apple Music was so good they stopped backing up only to learn months or years later that some tracks they wanted were substituted for ones close like non-explicit versions, etc.. The cloud isn’t your backup plan, it just makes it easier to sync. A cloud database also can get deleted or corrupted so test your backups from time to time.