Yehudis, I have a hard time believing you’d look at things that way if you grew up in a high crime neighborhood, or if you saw how violent crime and its perpetrators are glorified in those high crime, racially specific cultures. Mental health is a factor, but as i wrote, a person has a responsibility to get help before going psycho and hurting others. Failure to do so, short of being a halachik shoteh, which almost none of them are, doesn’t cut it.
In halacha, if someone is a murderer, and not a halachik shoteh, beis din couldn’t care less about his depression, ADHD, abusive past, or anything else. Maybe beis din shel ma’aleh takes those things into consideration, but it’s not the job of beis din shel mateh.
Re, avrohom and sdom – we don’t, as far as i know, have a source for davening for a rasha not to get punished. Avrohom was davening for an entire city, and we find the same concept by ninveh and yonah.
For individuals, we have a gemara with rebbe Meir and bruriah arguing about whether or not we should proactively daven that they die, with the conclusion that we should daven that they (other jews who sin) should do teshuva. The rishonim say that this applies to reshoim and not apikorsim, as we daven for their immediate deaths in shemoneh esrei “vechol haminim keregeh tovaydu.”