Well, whoever said that summoners led the armies. Both champions and summoners led the armies which for all we know could mean a champion is just as powerful as a summoner is. The main difference between the two is that summoners fight mainly using large scale magic attacks to eliminate many enemies at once whereas champions are more diverse in the way they fight but could very well be just as dangerous. And then there's champions who have received some sort of enhancement from a summoner before a battle, most likely making them stronger than a summoner in one on one combat or on par if they weren't already on par before.
Anywayyy. The purpose of champions fighting without summoners is just like why they would fight with summoners. They fought for any reason that gave enough reason to start a fight. It had nothing to do with summoners--they were just tools used in those fights.
Some champions were found and drafted (mainly uncontrollable threats to the general populace that no particular faction controlled ie. Cho Gath) while others became recognized as powerful warriors and joined willingly so that they could represent their side as a champion.
The champions are chosen by their city-states summoners most likely and the ones with no allegiance to any particular side are probably up for grabs for any summoner. Basically, Ezreal aside, champions are aware of when a League match will be occuring and probably discuss who should be chosen with their summoners. Then those chosen just wait to be summoned.
I don't know if they are always 5v5, but I'd assume all League conflicts are resolved in that form. Then again, TF and CS exist. Maybe both sides of the match agree upon which arena they wish to have their fight on?
The rules are made by the IoW. Most likely by the eldest summoners of them.
There are no more wars because the IoW either prevents them from happening altogether or stops them early on. And this generally never happens because noone is foolish enough to start a war with the IoW around. But whose to say they couldn't assassinate the most powerful summoners of the IoW and then take them down altogether? Or have multiple factions form temporary alliances to oppose the IoW?
There's definitely more than enough room for further exploration into this lore--and the IoW isn't stopping any of it.