Can a person have multiple mental health diagnosis?

Yes. many people have more than 1 psychiatric diagnosis.
Yes. Common examples of co-occurring disorders include the combinations of major depression with anxiety, alcohol addiction with panic disorder, alcoholism and polydrug addiction with schizophrenia, and borderline personality disorder with episodic polydrug abuse. Some patients have more than two disorders. Psychiatric disorders most prevalent among dually diagnosed patients include mood disorder.
MH diagnoses. Sure, depending on evaluation by a mental health professional. Comorbidities are common, such as depression with anxiety disorders.
Yes. It is actually quite common for certain mental health problems to overlap. Especially anxiety disorders and depression and/or depression and alcohol dependence. When two illnesses overlap it is called "co-morbidity". Fortunately some medications can treat both anxiety and depression at the same time. Many anti-depressants treat both anxiety and depression.