How to apply for a PhD scholarship in USA if I'm European?
In general, the process is to apply for admission to PhD program at a particular university and ask for financial support in the application.  In most STEM fields in the US, nearly all PhD students do receive financial support in the form of a 20 hour per week teaching or research assistant job.  

To answer your specific questions:


Application procedures vary from department to department, so you'll have to check out the web sites of programs that you're interested in to learn about the application process.
No, there aren't spots reserved for ethnic minorities.  In any case, as a foreign student, not a citizen or permament resident of the US, you wouldn't be classified as a minority student for most purposes.  
Financial aid in the form of teaching and research assistantships is nearly universal for PhD students in the US.