SONGS is a pretty sloppy heading for what are technically TRACKS, I agree, but to have that changed you'll have to ask Apple nicely.
I won't be drawn into a discussion about artists-vs-musicians, but (very broadly) Album Artist is the guy who issued the album and Artist is the guy who performs on a specific track.
Genre is free and open for you to use precisly as you will. I use the genre field widely (and often loosely) to help me find my way around my music collection. For example "African - Kenya" or "French 50's Pop". If you don't like what's there when you purchase a track, change it. (You notice that I'm also avoiding your "classical" trap).