What is the ERC (European Research Council)? Find more information here:

-ERC website
-Euresearch: the Swiss guide to European research & innovation

Q: Can the PI include PhD students in his/her project, even if he/she is not a professor and thus cannot formally supervise them?

Yes. In this case, an entitled professor has to officially supervise the PhD thesis of the students (according to the university’s rules). Please note that the PI can include in his/her project researchers of any rank (e.g., professors, PhD students, technicians) as long as his/her independence is guaranteed.