FIFA has complete control over all footballing activities in the world since it is the recognized regulatory body for the sport of football. Every sport has its own international regulatory body. For example: International Tennis Federation (ITF) for tennis, International Association of Athletics Federations (IAAF) for athletics, etc. These regulatory(or governing) federations become official only after receiving recognition as the official governing bodies for the sport by the International Olympic Council (IOC).
As the official governing body for football, FIFA has the following roles and responsibilities:
- FIFA has the duty to maintain top notch standards in soccer.
- It is tasked with administering the world soccer associations.
- It is in charge of organizing and promoting the tournaments.
- It is in charge of governing field matters relating to soccer.
- The body is responsible for constant development of soccer.
- It relies on the statutes to govern the system.
- It is also in charge of the Laws of the Game which are maintained by International Football Association Board (IFAB).
In answer to your questions:
FIFA has no powers like any judiciary body of a sovereign nation. FIFA can only govern in matters relating to the sport of football. Hence any trials, fines and bans will be in direct relation to the sport of football or footballing activities. FIFA's 'power' over private individuals extends only to their participation in footballing activities, including but not restricted to, playing the game or managing a team in any competitive game and taking part in any form of governance of football related activities.
Being banned from football means the person is not allowed to take part in any activities pertaining to football, i.e., they cannot hold any positions in football governing bodies, and they cannot be part of a team(player or manager) in any competitive match. They can be allowed to watch football games, if they purchase the tickets privately. Some people might not be allowed to even do that, which means they have been given a 'stadium ban'.
As to the corruption, well nobody can do anything about it, except the Ethics Committee in FIFA or the Court for Arbitration in Sport(CAS). Expecting the clubs to do something about FIFA's corruption is like expecting a player to show a red card to a referee in a football match. There is no chance of that scenario occurring whatsoever (even if it does happen, it will only result in the player getting into a lot of trouble). The maximum clubs can do is file a case against FIFA with CAS, although I don't see that happening, since none of the clubs have been directly affected by the corruption that has currently put FIFA in the limelight.
FIFA was recognized as the world governing body for football by the first few nations to play the sport in the early 1900's. FIFA's position became official upon recognition by the IOC as the the governing body for football in the world.
Asking football to continue without footballFIFA is like asking for a match to be played without a referee. Apart from the fact that FIFA governs football all over the world, FIFA also organizes tournaments like the World Cup, Youth World Cup, etc and has other roles and responsibilities as described above. A solution to what you are asking would be to disband FIFA and create another regulatory body for football. But the problem here is not FIFA, but the people running FIFA. Hence, even disbanding FIFA would make no sense if the same people are running the new governing body. The real solution is to get rid of the corrupt people, and ensure FIFA is being run in a clean and transparent way. Hence, FIFA President Sepp Blatter and FIFA Vice-President Michel Platini have been given 8 year bans upon being found guilty of corruption by the FIFA Ethics Committee.