Skills based on weapons (such as one-handed) level faster based on damage dealt, right?

Well, sneak attacks deal a lot more damage than normal attacks, as do power attacks.

Do either of them level the respective skill faster than normal attacks?

Power attacks definitely do. Not so sure about sneak attacks although thy do level your sneak I know.

Not sure but I do know that backstabbing increases sneak.

One of my friends is sure that sneak attacks were levelling his archery faster than normal attacks. Can't say I've noticed a difference myself.

It used to be the number of times you perform an action related to a skill, but in Skyrim it's the magnitude that counts. So if you perform a sneak attack, then you deal more damage, which increases your archery or one/two handed skills faster.

According to the wikia article http://elderscrolls.wikia.com/wiki/One-Handed experience gain depends only on the base damage of the weapon. Not on the smithing improvement. Not on the fortify one-handed enchantments. Not on the damage boost due to player skill level. Not on smithing.

It only depends on the number of times you use the weapon and the base weapon damage. So, a daedric war axe would give you best experience gain per hit. However, swords and daggers swing much faster. So the best weapon to use to improve one-handed skill is an unsmithed daedric dagger, which has 14.3 effective dps, followed by daedric sword. I would dual wield 2 unimprove daedric daggers and do regular 2 handed attacks for best outcome.

Sneak attack gives you 3x, 6x, 15x, or 30x damage, and a huge boost to sneak experience. But it gives you the same amount of one-handed experience gain for one-handed as doing one regular dagger hit.

My interpretation of the article is that power attack does not give you higher experience gain per hit.

I reckon it actually slows the progress, cause you hit less to kill. I wish it was based on how much damage you do, but then again, that would lvl sneak characters way too quickly.

Skill increase is based on damage done, so yes I would think so. However, I think sneak attacks might only affect the sneak skill, but don't hold me to that as I am not entirely sure.

I've found that sneak attacks actually slow your weapon leveling. It seems like the weapon's skill only goes up as if it was one hit.. while sneak skyrockets. When I play sneaky characters I focus on not being detected, and I find that my sneak is leveling many times faster than my weapon.