Does Scissors,Paper,Rock apply to League's roles?

Sort of, it's a long explanation but here's my best attempt at a summary.

First off, skill > counter picks. If a bronze V counter picks a Diamond they will more than likely, still get pooped on. Not because they were at a class disadvantage, but because the diamond has more experience and is over all a better player.

Now, moving on to the actual rock/paper/scisors of things.

In general, the way I see it Tanks/Bruisers>Assassins>Mages>Marksman>Tanks/Bruisers

Keep in mind this is very generalized, more explanation below. (all of the following assumes equal skill level from all players in the match)

Tanks and bruisers have the highest raw stats early game, and therefore are very difficult for assassins to handle. Typically assassins need to get kills to get ahead, and if you are against a tank as an assassin, you just wont be able to put out the numbers to do it.

Assassins have a great deal of burst damage, and mages have some of the lowest hp in the game, probably THE lowest actually. So, (especially with ignite) you will see a lot of people pick assassins into mages who are not tanks, for the first blood that they can more than likely get if the mage is not very very careful in the matchup.

Mages beat marksman early because they do more damage early than the marksman does. Not to mention that for poking someone down before going in for the kill, mages are usually better equipped for this with a few notable exceptions for marksman. A lot of mages also have CC effects in their kit, while very few marksmen do. So a mage can blow their entire kit on a marksman, stun/root/knock them up and walk away without taking a single point of damage. If played correctly.

Marksman beat tanks/bruisers for a few reasons. They are harder to get to without a really good engage tool, so for example, wont be able to easily get to an ADC early game. A lot of adc's have mobility built into their kit, and so they have even more ability to get out of the tanks range while still able to harass the tank from a distance. Late game marksmen will also typically build armor pen items that make a tank a lot weaker, so they can shred through them easier.

I would say no. Even in cases where there is a REALLY strong counter pick to a champ, you often have to kind of know how to play it correctly to make it very effective.

You can make an argument that the different classes do have a certain pecking order, but it often comes down to individual champs, and that all get complicated by the fact that you are really in a 5v5, not a 1v1. So I would say that if you're thinking of this game as rock-paper-scissors and trying to understand how to pick your champ based on perturbations of that theory, you're likely to mess up an awful lot.

Tanks/bruisers counter everything in this game. Whoevers team has more of those wins.

Not really, can go either way most of the time. You can say it depends on the matchup, but League is not as matchup dependent as games like DOTA.

You cant one trick and always pick Rock in DOTA, because some will actually just pick paper and you basically just lose. From my understanding.

There is a rock, paper, scissors, for some roles and it goes as follows: adc's counter tanks, assassins counter adc's, tanks counter assassins,

Mages and bruisers are generally more like jack of all trades.

you cant make a general statement like that based on roles.

there are mages that counter tanks for example.