So I confess to only having skimmed this but here's the deal as far as football is concerned:
World Cup and Euro Cup are every 4 years. They alternate, just like the Olympics. This year was the World Cup; in two years it will be the Euro Cup, etc. If we're using football (aka soccer) as a model for Quidditch, then the British Leagues should technically be playing fall to spring, rather than spring to fall. However, with the explosion of the Harpies Stadium, and the war, etc, I sort of feel like it's no big deal that this has shifted (when I came into the game there was talk of it starting up in the Spring, so I just went with it when I made that schedule). We can work on shifting this however, if there is a desire to.
No matter when the season is during our game, if we follow football, their British League Season would still directly coincide with the 4 year season of the national teams (hence the Irish National Team currently playing). In football, national teams hold training and tournaments throughout the Premiership and FA Cup seasons; players get pulled from training with their local teams in order to compete nationally. The seasons are all organized together too, so that they can pull players from any team and not disrupt actual matches.
I have a tournament set up for the top 8 teams at the end of our Quidditch Season, modeled after the FA Cup Final, which is actually different from the Premiership. Premiership winner will simply be the team that has the highest standing (which at the moment is Puddlemere). The winner of the 'FA Cup' (or Quidditch equivalent) will be the one that wins the playoff at the end of the season.
World Cup/national team matches would be completely separate. And as far as I know, Katie's the only player on a National Team at the moment.