Sorry noob question here. If I have example 35 points for one handed will it make me do more damage than if I had only 15? Or are those skills are just needed for perks?

Only thing I norice really is that my stats go up only when I find new better weapons. It seems I only find better weapons from mobs if I gained levels.

With 1-handed, 2-handed and archery, yes. Destruction, no.

Yes the higher the one-handed skill, the more damage you do. But it is quite small, that is why it can be unnoticeable.