I use 1 handed sword... but atm its quite slow and clunky... will I swing it faster the higher it levels up?

Naw, you do get a dragonshout that boosts your attack speed though. And if you dual wield then there's a 2 point perk for 35 % atk speed. (Quite an improvement)

To supplement the aforementioned, weapon speeds are in this order, from fastest to slowest:

Dagger
Sword
Axe
Mace
Greatsword
Battleaxe
Warhammer

I could've sworn you were wanting to play as a pacifist.

On topic tho, there's a dragon shout that will allow you to swing faster. I don't think level affects swing speed by itself.