What kinds of issues should go to my CSP partner vs directly to Microsoft support?

As a CSP customer, the general pattern is: your partner is your first point of contact for essentially everything — billing questions, licensing changes, technical issues, and platform incidents. They have the relationship and tools to engage Microsoft directly on your behalf when something requires deeper Microsoft involvement, so you generally don't need to (and often can't easily) open a direct Microsoft ticket yourself as a CSP customer.