For number 2, you should never try to contact an inmates family... Because inmate phones and mail are monitored, inmates sometimes try use staff to communicate with people outside the prison, but this isn't allowed. If, for some reason, they can't contact family/friends themselves, they need to discuss that with their unit counselor.
Prisons do have mental health services. In the prison I worked at, if an inmate threatened suicide during a weekday while mental health was there, they handled it. On nights and weekends, a nurse would talk with the patient and then contact the on-call mental health professional and explain the situation, and the MHP decided how to proceed. Usually, the MPH would speak to the inmate over the phone, sometimes "contract for safety" (meaning the inmate would agree not to hurt himself), and then mental health would follow up in-person with the inmate the next day. If the MHP thought the inmate was an immediate risk and couldn't wait for an in-person follow up, they contacted the on-call psychiatrist, and the psychiatrist gave a verbal order for the inmate to be placed in an observation cell.