If I upgrade my damage with perks in the One-Handed skill tree, does it also increase the damage of a conjured One-Handed sword?
It seems they are confirmed to benefit from the One-Handed skill: http://elderscrolls.wikia.com/wiki/One-Handed (first sentence under the One-Handed Skill heading):
The following are the confirmed weapons that benefit from this skill:
Sword - Iron, Ancient Nordic, Steel, Imperial, Blades , Silver, Dwarven, Orcish, Falmer, Elven, Glass, Ebony, Daedric, Bound Sword.
So with that in mind, I would only assume the perk also applies to it.