That only holds true for Israel, a theocracy, for whom those books of the Bible stand. What the rest of the governments do is up to them.
I’m for capital punishment, but God wants all people to come to Him. However, if we justifiably kill another, God will judge us in the appropriate context to that situation. If we murder someone, obviously He judges us in that context, too.
No, it is a Noahide rule...