I'm reeeeeally bad with drawing backgrounds, but I would be up for the challenge of drawing mystery dungeons.

I have never had my fire types "allergic" to water. XD I literally have a flareon swimming in one of the first chapters of a story of mine. I mean, it makes her feel uncomfortable and she's certainly not keen to do it again, but it doesn't harm her. Like what's been said, an open flame is different.

I like the idea of having more than 4 moves. I know that the 4 move limitation is just to make in-game battling fair, but it shouldn't apply to less confining settings such as a comic or a story. 6 seems alright, but I wouldn't mind having more. For my own 'circle' of stories and what have you, I set the limit at 8.

Badge teleportation is tricky. I too do not think it should be a get-out-of-jail-free card, but I also don't want to have the awkwardness of going to a 40f dungeon and rescuing someone on the 3rd floor and then being stuck there for the remainder of the 37f. Perhaps it only works out of range of enemies, such as in certain games where teleportation or fast travel is only available when you're 100% safe?