After you give the dragon scale from the Vyral to the natural philosopher on the windmill and he gives you a key, what do you do with the key?

Head to the Sochen Cave Palace. Enter the Destiny's March area from the east and then move around it clockwise, w/o entering the central area. When you get back to where you started you'll get a message that a door opened. Then you can just head through the center room to the west side and enter that door. Follow the path to the Hall of the Wroth God.

Be prepared for a long battle ;)

EDIT: You can ask further gameplay questions here (Thread 33821).

It is a long battle but it's oh so pretty. It's one of the most colorful beasts in the game.

Isn't the 'Wroth God' the Thunder Esper, Adrammelech? It would make sense, but I get the feeling I'm missing something...

Adrammelech is indeed surnamed "the Wroth." He's in Zertinan Caverns, though. If you want to dig up a connection in Norse or another mythology, be my guest. I'd love to hear it.

Nah I can't be bothered lol I'll just take it as a mistake on my part.

Where's the natural philosopher located?

Windmill #10 in Cerobi Steppe, specifically the Northsward section.