Hello, as a coach, if I'm elected president of the federation, won't that cause a problem?

Obviously not, since

1. it is possible to combine the two
2. it's the president who unilaterally chooses his coach, so it can be him.
   Of course, if you have two personalities in conflict with each other, there could be a problem, but because of the way the game is set up, there's no problem.

But you can't be president of federation X and coach of federation Y.