Clarifications on American graduate school for an European student
Speaking as a long-time graduate program director and adviser in a mid-level US department (at a big public university), I'd emphasize that most of our Ph.D. students in past years have had a less-than-adequate preparation in some basic areas.   This is true especially for US students coming from a typical 4-year college math major but is also true for some international students.   On the other hand, most of our students aim for a college teaching career, or perhaps a non-academic job, rather than aiming for a research career in some kind of mathematics (or statistics, part of our department).   At the most elite departments, it is taken for granted that everybody plans to do research.    But my own contemporaries in the 1960s Ph.D program at Yale actually went in many different directions.    

Like most other US departments, we have had an evolving procedure involving written (and perhaps oral) qualifying exams.    Anyone applying to one of the top few programs should certainly ask questions about how and when such exams are administered, and how they are evaluated. Typically no one wants to hold back a talented student, but sometimes a student overestimates his or her own talent and knowledge (and future job prospects).    Even Harvard and MIT Ph.D.'s sometimes end up teaching at small colleges or out-of-the-way universities.

In any case, identifying a potential thesis adviser (or two) is equally important, though obviously it's difficult to predict one's future interests precisely or to predict the future logistics of an active faculty member (sabbaticals and other leaves can upset plans as can personal crises).   Good luck navigating the US system, including the evolving immigration rules!