Yitz,
1. ?????? ??”? ??? ???????? ??? ????? ?????? ????? ???? ???? ?????? ??? ???? ?? ????. ??? ????? ??? ?????? ????? ????? ?????? ???? ???? ???? ??? ????? ????? ???? ????? ????? ????? ?????? ????? ????? ???? ???? ????? ?????? ???? ????? ???? ???? ?????? ???? ?????? ?????? ??. ??? ?? ????? ?? ??????? ???????? ?? ?? ???? ???? ??? ???? ?? ???? ?????? ???? ??????. ???? ?????? ????? ???? ????? ?? ???? ???? ????? (?????’ ??) ???? ????? ?? ????? ?? ?? ????? ???. ???????? ???? ????? ???? ??? ????? ????? .
Mitza (Lo Taaseh) 290 is that we should not reach a verdict by estiamtion even close to the truth such as where a pursuer chases someone into a house and the victim is found dying and his enemy is found over him with a bloody knife (this similar to the story of Shimon ben Shaatch in Sanhedrin 37b). The Sanhedrin shall not kill him on circmstantial evidence as there are not two witnesses who testify that they saw the killing as the Slomo HaMelech warns “do not kill one who innocent and righteous as I will not justify an evildoer” and the Mechilta says “and he pursues his fellow to kill him”.
Thus we see that Rambam is only referring to a bet din in a non-emergency proceeding (see Hilchot Sanhedrin 20:1 where he specifically says that this applies to a bet din).On the other hand, in Hilchot Sanhedrin 24:4 he says
?? ???? ??? ?????? ?? ????? ?????? ????? ?????? ?? ????? ?????? ???? ??? ????? ?? ???? ???? ??? ????? ???? ?????, ????? ?????? ??? ??? ????? ??? ???? ?? ??? ????? ????? ???? ??? ?? ????? ???
A bet din may flog one who is not liable to be flogged and kill one who is not liable to the death penalty, not to be over on Tora but to strengthen it, being they saw that people take it lightly. They should fence in and strengthen the matter as seems necessary to them? Rav Moshe (Iggerot Moshe Choshen Mishpat 2:68) extends this explicitly to a secular court.Form the story of David HaMelech and the ger Amaleki we see that the government may even execute a person on the sole basis of his confession despite what Rambam says in Hilchot Sanhedrin 18:6 (see Hilchot Melachim 3:9 with Or Sameach and Responsa Chatam Sofer Orech Chaim 208). For a wider discussion see ??? ????? ? ?? (in Hebrew) available on-line.
2. Regarding Graham, Harris County prosecutors filed an affidavit signed by the bailiff who escorted Graham from the courtroom after his death sentence, who heard him say, “Next time, I’m not going to leave any witnesses.” A prosecutor filed an affidavit stating that the bailiff related the comment to him within minutes of the time it was allegedly made.
Harris County District Attorney Johnny Holmes noted that Graham’s case was reviewed 35 times by the courts and that his conviction was never overturned. The Supreme Court rejected Graham’s appeal in May.
3.Prosecutors have been disbarred for misconduct.The general (but not absolute) immunity is necessary as otherwise nobody would want to be a prosecutor (as in Illinois nobody wanted to be a gyneclogist until the law was changed). Chazal also discuss various immunities that were given for this purpose. A cynic would say that their misconduct is balanced by defense misconduct.
4. I also went to law school and took law courses in HS and college.