'Skins' are alternate costumes that a Champion wears at the urging of the Champion's Summoner. They're usually for aesthetics, but some Summoners seem to like using skins as a way of indicating they spend more time working with a certain Champion. It might also make some Summoners feel superior to others, and may lead them to a false sense of entitlement to the chosen Champion (i.e., "Darius or feed").

Why do enemy Champions never notice Teemo if he stands still long enough, even if they basically run him over in the process?

Actually, Teemo is part ghost.

Why is the cinematic so different from the gameplay.

Everytime Teemo stops moving he is immediately begins to grab the equipment from his pack to disguise himself as part of the scenery, hence the name of his passive "camouflage" because Teemo has already scouted the area and has a good understanding of lay out he is adapt at quickly and efficiently camouflaging in with the environment. Other champions who are not as observant of the land and have not properly scouted the area will not make notice of the small changes that Teemo has done simply walk right past him.

Teemo also has an excellent understanding the ridged terrain or land pathing and as such will make it so while it seems Teemo is being stepped on by say, a Cho Gath, Teemo is in fact no more than a few inches away when from the hulking Void Terror.

Why does Graves Buckshot ability fire 3 shots but his shotgun only had 2 barrels, As well as Rumble with his ultimate firing 6 rockets on the ground and him only have 3 on his mech?

Well you see, The bullets graves use for buckshot are different. They split in to 3 the moment they are shot out of his hex tech gun. Rumble lunching 6 but he has 3 missiles beats me.

What kind of bait does cait use for her Yordle snap-traps. Kinda look like cupcakes to me.?

Yordle Snap Traps emit a sound heard on a frequency so high, it can only be heard by Yordles. This sound insults the person's mother and causes the unsuspecting Yordle to attack the trap by stepping on it, causing it to become trapped. For non-Yordle creatures, there is bacon in the snap trap, and everybody likes bacon.

How does Viktor's invisible laser work?

When Viktor fires his laser, it is always visible. However, since it is highly advanced, maybe even past pulsefire ezreal, it occasionally will not be detected by summoner magic. It still damages because it hits, just summoners can't see it.

Why can Rengar insta-stealth?

Rengar is an amazingly accomplished hunter. He has only been bested by one creature (that we know of) in almost all of memory. Rengar's extremely advanced hunting skills also include the art of camouflaging himself, and with his extensive time spent hunting, he has developed this skill to be almost a forgettable action, as easy for his brain to think about, signal, and activate his body as if it were breathing or his heart beating. It just happens. The process occurs somewhat like the following scenario:
Rengar sees his prey
He estimates relative distance
Decides what moment to become completely "invisible" (just so expertly camouflaged we do not comprehend it) (perhaps a better way to explain it is what moment to become extremely cautious)
Begins process of grabbing natural fauna to attach to his person
and by the time he would be normally seen by the enemy, he is "invisible," and therefore undetected.

Unless of course a vision ward gives the enemy unnatural sight.

Speaking of which, how does the vision (pink) ward give such excellent vision? Is there a scientific explanation or is it pure magic created by the Institute of War? Or some other reason?

Pink wards have advanced technology sensors that scan an area, providing total vision of any moving objects, stealthed or not.

If you run into someone as hecarim without E up, how does it not do damage by itself?

It helps him do damage.

How does people like Ashe stand such cold temperatures in her homeland if she is wearing slightly skimpy clothing.

Ashe, Tryndamere, and Sejuani are all from Freljord and all dressed inappropriately for the cold weather. This is because the people of Freljord have developed higher body temperatures to protect them from the cold. Because their body temperatures are so high, they must wear less clothing when they go to warmer places like Summoner's Rift.

Why do Ezreal's abilities increase his attack speed?