How are mathematics PhDs from the USA viewed in mainland European universities?
Do American-educated mathematics PhDs generally have an advantage over natively educated PhDs (in, say, France or Italy) in applying for research positions?


Generally speaking, no: why should they? And for what concerns specifically Italy, take into account that some things that can make an application strong(er) in the US (e.g. stellar recommendation letters or awards), might go totally ignored by a selection committee in Italy. This means that you should tailor your application to the country you're applying to. Let's see why.

A selection process in Italy, both for permanent and non-permanent positions, is usually composed of one to three steps, where each step counts for a fraction of the final mark (details vary across fields and position levels):


CV and publications (this step can count from roughly 30% to 100% of the final mark). The selection committee, before knowing the candidates, decides how to evaluate each section of the CV. If, e.g., the call doesn't explicitly mention recommendation letters, these are typically discarded; if the committee decides that awards count at most 2 points out of, say, 30, even if you have a Nobel prize, they (probably) won't add more than two points (should they do, in case of appeal from some of the candidates, the call can be invalidated).
One or two technical essays (optional; when present can count from roughly 30% to 50% of the final mark). Here, the applicants are typically required to write a technical essay on a specific topic in the field of the call. I don't know about mathematics, but in my field one might be required to describe how a certain instrument works (e.g. a spectrum analyzer) or how to perform a certain experiment (e.g. how to measure the fine structure constant). The difficulty of the question depends on the level of the position.
Interview (optional, depending on the position). Here the applicants are typically asked to discuss technical topics in their field or, for instance, some of their publications. For an associate- or full-professor position, this step can be substituted by a (public) lecture on a specific topic in front of the selection committee.