What is the ruling on insulting a Christian, a Jew, or an idolater and accusing them of being illegitimate, and does the Hadd punishment for slander apply to such a case, even if they have insulted the true religion and slandered the Prophet?

The prescribed punishment for slander (hadd al-qadhf) is not applied to someone who slanders a disbeliever, because this punishment is specific to slandering chaste, believing women. However, it does necessitate a discretionary punishment (ta'zir).

If a disbeliever insults the religion or slanders the Prophet, peace and blessings be upon him, then he is considered to have violated his covenant and deserves to be killed, according to some scholars. This is to be implemented by Islamic courts and not by individuals.