Can I track my workout and listen to music with Apple Watch Series 2 without having to carry my iPhone?
You are correct on all three counts.

The Workout app will work without your iPhone, at least. Third-party apps might need to be updated to be completely independent. Strava, for instance, won't start without an active iPhone connection, but Runkeeper will.

Both the first generation WATCH and Series 2 support syncing of a single playlist up to either 2GB or 250 songs.