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?

2 Answers

It seems they are confirmed to benefit from the One-Handed skill: (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.

Yes. Your conjuration skill will increase when you create the sword but after that it's treated like a normal sword.