Can anyone do a psych exam?

Yes and no.: There is nothing to prevent anyone with an interest from learning the basics of a pscyh exam. Most of the questions are straightforward. However, learning to interpret the results is why people spend years in training. There are lots of screening tools for all sorts of mental disorders, but they are just screens. If concerned, work with a professional.

Nope: Only a licensed mental health professional can do one. Usually they are performed by psychologists or psychiatrists. And it can't be a friend or family member, even if they are a licensed mental health professional.