So I played over 15 hours of the demo and I think I explored everything.
1. It rains, but only when you're going to Insomnia
2. There 3 dungeons
First. Boss is the spider Arachne
Second. The elevator (showed also at PAX), boss is the samurai
Third. A ladder down to the dungeon
3. Found a tunnel where Goblins are
4. You can climb a ladder of a tower in the north between the tomb and dungeon
5. You can go to the place where Noct and co are looking to Insomnia and see the Niflheim airships while Noctis is calling Cor