Aspergers is a form of autism.
The terms mental health or mental illness are poor but I would only include bipolar, schitzophrenia, and psychosis. Developmental delay and autism are separate.
So you should enter no for mental illness for yourself and could do the same for a family history of mental illness but you should add family history of autism at the end. Any genetic element to autism is poorly understood, but that is irrelevant as if you had it you would have had it from birth. You aren't going to develop it in the middle of a flight so you are not a risk.
If questioned you simply answer no to any question about whether you have had autism, developmental delay or problems at school. Volunteering that someone sometime MAY have speculated is not required unless a doctor told you that they were considering the diagnosis. If this is the case, ask your GP to confirm the hypothesis was refuted in a one line letter.
I would be very surprised if you have a problem, but you must be accurate with your statements and exclude hearsay