How does the American PhD system works for foreign students?
In most fields, US PhDs can start right after a bachelor's degree. Some incoming US students will have masters degrees, either because they've been uncertain about their career path and or because they pursue a masters to get more experience to make their applications look better (especially if their undergraduate research experience is lacking).
Because US PhDs usually start right after a BS/BA, they are typically longer than the European variety; 5 years is common, though for some fields they can be much longer, up to 10 years.
Coming in with a European masters (or a US masters, for that matter), there is not typically anything that will allow you to complete the US PhD faster than normal. There might be cases where you can skip some coursework, but that will be handled on a case-by-case basis.
Funding can sometimes be more difficult for foreign students unless their own government provides support, since some funding sources are not available to international students (especially fellowships from the US government). Other funding is available to all (most TAs, sometimes with a more strict language requirement, many RAs and PAs; check with programs you want to apply to about their funding availability).
Many programs and universities will have information for potential incoming international graduate students, you should search the websites for individual schools/programs you are interested in. Some of this additional information will cover things like visa requirements (including funding) and language proficiency. An example from my own institution is here: https://grad.wisc.edu/international-students/
As far as PhDs as a job, I'm less familiar with Italy but have seen the German system explained often here: US PhDs are not jobs like German ones are. That said, almost all US PhD students (and especially in STEM) have some sort of funded position that includes a stipend, and typically this comes with tuition remission, so you take home a (modest) income to pay expenses, while not paying tuition. Other students are on fellowships or traineeships that specifically fund a students' education and include a stipend.
For some fields, these are usually Research Assistant positions, where you work on a research project for pay (sometimes this is the same as your PhD thesis, sometimes not - again, depends on field and circumstance), or Teaching Assistant positions, where you teach or help teach a course or two. Programs typically guarantee some sort of funding for X years; if you take longer, you could possibly find yourself without funding (the likelihood of this depends on the field; I don't know about theoretical physics but my educated guess would be that there are enough TA positions in physics that as long as you're willing to teach you'd find a TA spot). There are also cases where someone might do an unfunded/self-funded PhD - unless you (or your family) are independently wealthy I would strongly advise against this (you will find other Q&amp;A here about it).