Yitz,
1. The Ran says (Derasha 11) that hakpada on Tora evidentiary will lead to anarchy. Therefore , it is necessary to establish a government that will enact those laws that it deems necessary to keep public order. In fact, it can even impose the death penalty for minor offenses such as not going to where he is told to go or violating house arrest if he does so out of rebellion(Rambam Hilchot Melachim 3:5).
2. It is true that felony murder is not murder under Tora law but the government certainly has the right to consider it murder (ibid Halacha 10). The example of fire, while it is only a civil liability under Tora law, is, in my opinion, apt.Damaging by fire is liek damaging with one’s own hands. The fact that it is not a capital offense if he did not intend to murder (although he may well be considered close to meizid and thus he is thrown to the mercy of the goel hadam – and there is an opinion that a government officer has the din of goel hadam).
3. I disagree with your definition of ??? ????? ????? . It is certainly not the peshat. David HaMelech had both the ger Amaleki and two brothers who admitted killing Ish Boshet summarily executed (not even a trial) on their confession alone (Shmuel Bet 4:13) despite what Rambam says in Hilchot Sanhedrin 18:6 that we do not punish people on their confessions (not even malkot) as this may be the way out for someone who wants to die to commit suicide.
4. Even under Tora standards an innocent man might be executed. Navot was, in fact, executed according to the testimony of two witnesses. Ramban says (Devarim 19:19) that if Hashem allowed it the defendant must have been liable to the death penalty for something. In the case of Gary Graham that you once cited, as a Noahide he was certainly liable to the death penalty for his various crimes. This, of course, is not our cheshbon, but it seems that the POSSIBLY innocent men executed were guilty of various crimes for which they should have received the death penalty.