Varies by County. Some counties dish out permits with little hassle or BS. Other counties put applicants through interrogations with judges, etc. Most (if not all) NYS Counties require as many as 4 references to "vouch" for the applicant's character during the application process.
I think you have the wrong idea, it not an exam, it is a check. They check and see if you have an history of mental health problems. This check would be requested by a county and Albany would send the info back. Albany has been backed up, and some counties are now doing these checks themselves, and saving time.
As for the interview, I had to do one. But I think it depends on the county.