What songs should I use to represent these genres of music?

Yeah for thrash metal "Don't Tread On Me" doesn't really fit since that song isn't really even thrash, it's more just basic heavy metal. I'd suggest "Holy Wars... The Punishment Due" by Megadeth. The song is the pinnacle of thrash metal. Or if you wanted to use Metallica there are plenty of better songs off their earlier albums like "Blackened", "Fight Fire with Fire", and "Battery" to name a few.