I would think no one would want someone who is mentally ill, suicidal, on large quantities of antipsychotic drugs to own a firearm. Under HIPAA laws isn't it illegal for anyone to possess this information? If someone is truly mentally ill, and medicated how would that show up anywhere? Does anybody know?

I think background checks need to be more extensive. The background checks they have in effect now are obviously not working. I also think the age should be raised to 21, to own a gun. Assault Rifles and bump stocks need to be banned for civilian use.

Who will define responsible?
How would you determine responsibility?
The term is very subjective.
Here in Ohio, we used to have to complete a training course provided by a state licensed instructor. But even that approach was only an attempt to insure competency in the use of a sidearm, and knowledge of the laws and regulations concerning it's use. Then, there is the age requirement. A standard sign or measure of maturity/responsibility.

I believe the premise has always been "responsible until proven irresponsible by actions".
Much like "innocent until proven guilty."

How do we know someone is responsible enough to drive a car?

A background check isn't going to tell you if a person is responsible it is only going to let you know if they have had any previous run ins with the law.

No.