What I mean by animation-skipping is pressing a confirm button while animations like battle animations/text, overworld animations and screen transitions, etc. are playing to move forwards into the next queued action. And yeah, running being too slow, combined with the lack of diagonal movement in diagonally-oriented maps, makes getting around quite a chore, especially if you want us to backtrack to investigate. If you don't want to trivialize running away from enemies, having some diagonal movement that's deliberately slow would still help, as would a faster running speed when there's no enemies on the screen.
Yes, it's a very pretty game! I would like to return to it again.