If someone is mentally ill to the point they are considered dangerous and shouldn't possess a firearm, then they are considered dangerous. As in period. As in, they shouldn't be free to harm someone with a gun, or a bat, or a hatchet, or an icicle, etc ad infinitum.
If they are a danger, then they are a danger.
Yes and no. Profound mental illness is a very strange beast. For quite a few years, I dealt with several people with schizoaffective disorder. On proper meds, they were very quirky but otherwise boringly ordinary. Off the meds, they ran the range of symptoms from bipolar/manic to delusions to hallucinations to full-on psychosis. Should they be allowed around guns? Hell no. Should they lose their freedom because of their illness? In most cases, hell no.