As an HR professional, I can tell you that the employer can ask if your illness and time away is due to an FMLA covered illness. Even if you say no, they can designate your time away as FMLA. It is in your best interest to be covered by FMLA as it protects you from losing your job due to your illness. That said, you don't have to put a diagnosis on the FMLA paper work. They can direct you to fill it out and if you don't that it considered insubordination and they can discipline you.

I'm not sure how to answer your question about work related issues since I'm not sure which ones you are talking about. Whether or not to tell your employer is a big decision. If you are having symptoms that are majorly affecting your work, you may have to. You can ask for accommodations under ADA and also get more time off than FMLA covers. Otherwise, you may not want to tell them as you don't know what prejudices people have about mental illness and it will affect how they see you and treat you.