but I think BG3 will put limitations on unarmed movement and there won't be any wall-running or water-running shenanigans in the game.
I absolutely expect Larian to give us a replacement for wall-running that helps with vertical traversal. Think permanent effects of the Jump spell, jumping distance derived from Dex or something like that.
Slow Fall should work with armor, shouldn't it?
Actually, you're right. I had to double check that. It makes little sense, but hey, it's RAW.
Also, I had a small brain fart tbh, shield proficiency would not be available to gith monk, so my best bet is going for the greatsword aka silver sword for flavor. As I've written above, I wanted to combine two desired tropes in a single run, because my time would be short after the game releases.
Hmm... have you considered taking your 1st level in fighter and then switching to monk? You'd have delayed extra attack and miss out on level 12 ASI but no actual monk features. As a benefit you'd have heavy armor, shield and all martial weapon proficiencies upfront, which would free you up to round up your Strength to 18 with Heavy Armor Master at 4th level. Also you'd get dialogue options for two classes.