There is a type of murder that involves the death penalty.
When you say rape, do you have a definition? One might be needed to differentiate or clarify matters of law. Rape is not seen by all to be a reason for or crime that involves the death penalty. But I am wondering if the rape victim is engaged or married, and if she cried out in the city or not.
Deuteronomy 24:7 NASB - 7 "If a man is caught kidnapping any of his countrymen of the sons of Israel, and he deals with him violently or sells him, then that thief shall die; so you shall purge the evil from among you.
and lying in court about a capital crime.
Lying in court and bearing false witness. Is this relationship the one that you see?
Execution method should be painful, carried out as soon as possible after the trial (which should take a few days at most) and broadcast live. Stones are cheap and easy.
Anything that you have in mind right now?
Deuteronomy 19:15 NASB - 15 "A single witness shall not rise up against a man on account of any iniquity or any sin which he has committed; on the evidence of two or three witnesses a matter shall be confirmed.
Stoning is not always the preferred method, but if you are doing things according to God's Law it may not be the method prescribed. You are eager unto execution. Is this your enthusiasm for the Law? You might want to learn about the law's intent, but mercy is not always the right way. You are taking the matter head on as you feel you should.