I'm afraid rule of cool overrides the need to put a meticulous, scientifically constructed and internally consistent magic system in place. Also not being given an explanation is not the same as "an explanation does not exist". Take the magic with a fair amount of suspension of disbelief. You don't need everything explained to know that it can happen.
It is possible to make a game that is balanced and internally consistent with a deep, rich lore that's been well thought out and meticulously designed with the creative detail of the Lord of the Rings or A Song of Ice and Fire, but currently in League of Legends that's immaterial to its present business model and its genre. Lore and story is not what makes most of its money, an entertaining game where all manner of players can come to test their skills in a team-based battle for an objective is what makes most of the money.
If they get a cool idea for a new champion, they're not even going to want a lore team to tell them "You can't do that, it contradicts X, Y, and Z." Besides, any creative mind can get around the perceived holes. The holes you bring up aren't even "This is wrong", they're questions that you just want answered. To be honest, you're grasping at threads and making me wonder how seriously you really are about this. Stuff like "Vi is a fierce fighter determined to punch criminals in the face. Why would she punch Lulu, Annie, or Amumu in the face?" or "Wouldn't Thresh answer to Mordekaiser's commands on the field even if on the opposing team even with a summoner merged with his mind?" just makes me hit my head on my desk.
They're participating in an arena based competition in a league called the League of Legends. This is what they came to do, or what they MUST do in some cases like Nocturne. It is, BY LORE, essentially a competitive sport, though often with much higher stakes than what a modern competitive sport would have, but like any competitive sport it has rules, it has referees (the Institute of War), it is not a mindless battleground, or a war. So the questions like "Why would she punch someone like Annie in the face?!" or "Why doesn't Mordekaiser tell Thresh to do what he wants so he can win?!" are pretty damn easy to explain. Why? Well what happens in real world sports? You either get benched or disqualified because it's unsportsmanlike, or worse if it's conspiring to throw a game. If I played defensive line for one team and my little brother played quarterback for another team and our teams were to have a contest I would be obliged to plaster my brother across the field if I had the opportunity. By your analysis this would constitute a lore problem, because why would I hurt my own brother? Anyone with a lick of sports sense should know the answer to that.
Hell, read Leona's lore and ask why she could attack or even kill Annie. Answer: She probably doesn't enjoy it, but Annie certainly isn't going to pull any punches, and it is nonetheless the expectation. Annie certainly doesn't seem to have any problems with this, and while that may be a sign of her questionable mental sanity, that's something that needs to be looked at outside of a match. Acting out in a match probably isn't going to turn out well, given the Institute of War, by lore, is REALLY INSANELY FREAKING POWERFUL. Presumably all champions accepted this when they went in. If they were to suddenly question it and disrupt a League match themselves, it would likely cause quite a stir and get themselves suspended. Many of them are there to represent what they stand for, be it strength, glory, honor, or something else.
Key point in all this: Institute of War, really insanely powerful. Champions can do some pretty impressive stuff. Some can screw up vision, taunts, charms, etc. Chances are that the summoner does a fair bit more on his or her own that isn't related to gameplay.
Essentially you're asking what are the rules of the League of Legends that the engine attempts to enforce? Obviously it grants quite some freedom, but there are likely various rules that still apply, but apply equally to all champions, unlike the incident of Jax getting special restrictions.
Did I also mention money? Hiring a big lore team is a big money sink to appease people with. Were this an RPG, yeah, you really would need that kind of system, and the more thought out the better, but this is a MOBA. We're not playing Dungeons and Dragons here. The game itself can go as it has been. Lore can still exist outside of it.