Yes, PAX footage shows Noct and bro exploring a mine dungeon full of traps, monsters and boss.

Did you never played VII, VIII, or XIII ?

No, Tabata himself said that you can enjoy the game fully without any knowledge about the movie or anime. The animes are short and only delves into the relationship between characters and doesn’t really have impact to the story. The movie happens before the main game, and the events are mentioned in the main game…

Just like very other FF mainline title, you only get access to Regalia Airship Mode upgrade nearing end game. For most of the game, you use the car.