are itunes and apple music the same thing

No. Apple Music is a subscription service. See: https://www.apple.com/apple-music/

You can still use the Music app on an iPhone or on a Mac that is running Catalina or Big Sur to purchase iTunes music without having an Apple Music subscription. See: Buy music from the iTunes Store - Apple Support

Yes. The following information may help: Turn on Sync Library with Apple Music - Apple Support