"I heard some things, which are very different from where I live. Is it right that teachers can compete together with their students? Do they have to do that, or is it up to them? Do you have little couples consisting of two students, and teachers have to fill the gaps for missing dance partners? What kind of people do compete in America?"
There are a small handfull of amercians dancing somewhat as your Italian IDSF and professional couples do.
There are a larger number of semi-serious amateurs, who compete from beginner level ranging up to the same events as the IDSF couples.
But the largest aspect of american ballroom revolves around money. Rich students (and also those spending money they cannot afford) pay to dance with their teachers, for lessons, socials, and also possible for competitions which are the most expesnive of all. Many professionals do at least some work in this area of the business, because there is so much money to be made there. Some have dozens of low level students, others who are perhaps serious professional competitions may have only one moderately advanced student who pays for enough of their time to in practical terms be almost their sponsor.