This past year the Archdiocese of Washington allowed you to send HSP test scores to up to 4 schools plus one additional school in either Archdiocese of Baltimore or Arlington. You could also email the Archdiocese of Washington if you wanted to send to more Archdiocese of Washington schools (we did and got it sent to a 5th ADW school at a cost of $15). Each Archdiocese has its own approach. Check their website and take the HSPT with the Archdiocese that best fits the location of the schools you are applying.
https://adwcatholicschools.org/high-school/placement-tests/hspt-dates-locations-fees/
No quotas per se but I do think competitive schools overall try to get a diverse group of incoming schools so perhaps there is an informal cut off based on applicant pool.
You don't need straight As or strong HSPT scores to get into most of these Catholic high schools (exceptions are Prep, Abbey, and the like). They look at whole child profile. It helps if you play an instrument or have something else that stands out along with good recommendation letters etc.