Skins that affect game mechanics

Solution 1:

As mentioned, the only thing skins change is the appearence. No stats are affected nor supposed to be. If they were, people would fully classify lol to be a pay2win game, which few does at the moment.

One thing to note though, is that some skins can change the appearence of their spells. I've noticed some times that the abilities are actually harder to spot by the eye on certain skins.

An example is Pulsefire Ezreal, whom in my opinion is harder to track some of his abilities on. His skill shots gets a bit harder to see exactly where the hit area is.

I can aswell imagine Frostfire Annie and her abilities to be a bit more hidden during the winter version of Summoners Rift.

A clear difference is the Nocturne original and Nocturne halloween skin. The Nocturne halloween skin's Shadow ability almost cuts through your eyes due to its strong colors. It might appear to be a bad thing, but can be useful to drag attention with when you're going tanky with him.

But basicly, the only thing skins might improve your gameplay with is visual advantage to a low extent, and the feeling that you must win because, hey, you got a skin !

EDIT
It seems as if Singed's poison had been purple all along. Never mind this case:

For example Surfer Singed (used to) have an other color on his Poison Trail. It was purple, and it has harder to spot than the normal green one. I noticed enemies running into my purple poison sometimes because it was quite hidden in the brown ground. That is however altered now since they increased the visuals on his poison and made it green for all skins.

Solution 2:

Skins have absolutely no effect on your damage output/resistance, attack/projectile/movement speed, or on-hit effects.

In a nutshell, all a skin does is change the appearance of your champion. There are some cases (Legendary skins, for example) where the skin will have a more significant effect on your champion. This can come in the form of skills looking different while keeping their original effects, champion voices changing, or the way your character walks.

Some (but not all) examples:

• Frostfire Annie - Blue flames instead of orange.
• Gentleman Cho'gath - A more civilized English accent, as well as some altered dialogue
• Woad Ashe - Different movement animation (still maintaining default movement speed)

Currently, the only champion whose skin affects other champions (in addition to his own appearance) is Pulsefire Ezreal. However, even this is a minor change, changing the enemy champions'/minions' death animations.

The case that you mentioned regarding Sunglasses wearing champions in relation to Leona is a rare case. Off the top of my head, it is the only case (I could be wrong, though), and even then the damage reduction is utterly minuscule.

For additional reference, the Skin Spotlights channel on YouTube has a fairly comprehensive list of videos showing various and many champion skins.