Does the Bow used affect the damage the arrow does? All I can find is the skill affects damage, and nothing on the type of bow. So Longbow does the same amount of damage as an ebony bow if they both fire Elven arrows. Am I wrong?

Have you checked the UESP? My understanding is that they are independant of each other and you add bow damage to arrow damage to get total damage. So a longbow with ebony arrows will do more damage than a long bow with iron arrows but an ebony bow with ebony arrows will do more damage than a long bow with ebony arrows.

Archery Skill increases increase all bow's damage. Arrow damage stays the same (as far as I can tell)

Arrow damage is added to the Bow damage. Not sure if there is an additional multiplier or not.

So, yes, the bow matters, and the arrow matters too. Thing is, arrows, unless there is a multiplier, do not have enough of a damage range to really offset a weaker bow as compared to a stronger one. Instead, what I do is use lower level arrows on weaker enemies, and higher level ones on stronger enemies.

There used to be a multiplier, not anymore though.