How much of a difference does charging the bow completely make when using normal stone arrows? Are better quality bows the only way to do more damage with the bow and arrow, since the Melee Damage skill doesn't affect it when used as a ranged weapon?
In my experience charging a bow can significantly impact arrow damage.
The wiki lists that bows do full damage, 100%, when the are charged for at least 1.2 seconds, and do only 41.25% damage when fired as soon as possible, at 0.3 seconds.
For stone arrows this means doing 20.625 damage to 50 damage (with an engram quality bow).
Higher quality bows will increase the total damage, and you are correct no player stats will improve ranged. However, you can do more damage with headshots against most targets.