About the last point- I remember hearing somewhere, though it could be an urban legend, that that's precisely the reason early Dungeons & Dragons (OD&D/AD&D era) had so many variations within different dice roll mechanics for whether or not a high (or low) roll was good or bad (ie high rolls for your attack was good, high rolls for initiative were bad).

If the player used the same dice for all rolls, a balance check against biased or loaded die was therefore built directly into the game, with the perk of making it very obvious if a player was using specific dice for specific rolls