What kind of tests does the hospital run to confirm if a person is schizophrenic or bipolar, etc.?

Behaviors, not tests: A physical exam and/or lab tests may help a doctor rule out medical illness as the cause of some symptoms, but it is by observing a person and - perhaps psychological testing - that a diagnosis is made. Even though many wish it was different, with respect to mental illness, usually there is no clear confirmation - only expert opinion. And, they can be wrong without being culpable. It's difficult.