What is the historic basis for the hatzi kaddish service?

Answer 1

There is no "hatzi kaddish service." The Kaddish is a family of related prayers in the Jewish liturgy, all said in Aramaic. Because they are in Aramaic, we can date them to during or after the Babylonian Exile because that's when Aramaic seems to have become the language of the Jews. And, the kaddishes were established by the time of the Rabbinic sages (the Tannaim or Pharisees) whose sayings make up the Mishnah. Technically, these are doxologies, that is, short prayers of praise that are used to punctuate longer services, dividing or marking the ends of sections of a service. There is the long Kaddish, the short Kaddish (known as the chatzi Kaddish in Hebrew/Aramaic), the Kaddish after study, and the mourner's Kaddish. The short Kaddish is used at several points in the traditional liturgy, so there is clearly no single service it is associated with.