I've seen a few places where some of the champions are referenced as past summons. Whether this is confirmed or not, it's a curious concept. The theory of the league is to prevent massive disaster like the Rune Wars from being repeated, but who exactly are the summoners and what are the problems they are attempting to solve?

It's a bit open, but could our prince be a summoner one day and a champion the next depending on the conflict?

Yes, you can double-dip.

But both roles are highly specialized requiring professional excellence for either. You can't "jack of all trades, master of none" you must be master of both. So it is technically possible, but practically impossible for most people to do due to the time consumption of mastering two completely different forms.

confirmed in journal of justice.

The reason for the league existing is essentially to replace warfare with small-scale gladiatorial combat.

For example, nation A has a dispute with nation B over a plot of land. Nation A and nation B both send champions to compete.

Nation A's team wins. Nation A takes the land.

This is an alternative to a 3-5 year long (potentially longer) military invasion and occupation that would otherwise cost thousands (potentially hundreds of thousands to millions) of lives and billions of dollar-equivalent-currency to reach the same conclusion and results.

It's a shoddy system that is fraying at the edges because no nation will want to repeatedly suffer defeats and constantly have to give land to another nation based on the results of a couple gladiatorial matches, which is the result the league imposes.

Imagine if you live in, say, Ontario, Canada. Canada has a hockey game with England. Canada loses. Ontario now belongs to England and is subject to whatever laws they so desire to impose and treat the area however they wish, based on the outcome of that one hockey game. If you decide to stay in Ontario because your home is there or your business is there instead of becoming a relocated refugee, you are also subject to any laws and conditions England wishes to impose. If they decide you have to pay extra taxes for being Canadian, they can do that and according to the world as a whole it is entirely allowed and justified. That's a fairly light example.

Needless to say, not a system without flaw.

But the idea is that managing a nation becoming disgruntled from the results will be less destructive than England launching a full scale invasion of Canada and taking Ontario by military force, as both of these nations are nuclear powers.

That is the league's purpose. How effective it is at its job is subject to debate.