"many mass murderers on death row exhibit the longing to stay alive so they must have something functioning within them that can serve as the basis for empathy"
There's a whole world of difference between empathy and the inherent instinct for personal survival.
Do not make the mistake of imagining the sociopath's apparent empathy at any point derives from anything other than the calculated means to achieve his own ends.
Sociopathy is not a mental illness. It's usually defined as a personality disorder and is notoriously difficult to treat, essentially due to lack of personal insight and extreme egocentricity.
