I've noticed that there are varying minhagim among people whether to stand during Kaddish or not. Is there any actual halacha regarding this and whether it's required?
The Mishna Brura OC 56 sk 8 implies that one cannot sit down during kaddish, but if he was sitting already then he may continue to do so. However, he says it is proper to be strict and stand up anyway. I note that the whole standing up discussion is limited to the part of kaddish until "da'amiran be'alma ve'imru amen".
Great sourcing, Double AA! Just translating them here... OC 56:1 has the Bet Yosef (followed as the great decisor by Sephardic communities) not enunciating a preference for standing during Kaddish. The Rema (Ashkenazic custom) then interjects that "it is proper for one to stand when they answer Kaddish & all matters of [special] Holiness." Be'er Hatav (ibid, note 4), following the Turei Zahav, expounds that "the Darkei Moshe (also from the Rema) writes that the Mahari"l would not stand up for Kaddish, however any Kaddish that would find him standing, he would remain standing until after 'Amen! Yehei Shmei Rabbah'. And so too wrote the Sheyarei Kenesset HaGedolah, 'I too follow the custom of the Maharil.' View OC 55. And so too was written in the Sefer HaKavanot of the Arizal, and this is its language there: 'This that seems to be indicated from the Jerusalem Talmud that one must stand for the response of Amen, Yehei Shmei Rabbah is an error. For this is not from the Jerusalem Talmud itself, merely that one wise man [quoted therein] would do so.' And in the Kavanot, there is a gloss, and this is it's language: 'What my Master za"l was accustomed to doing was that any Kaddish that followed an Amidah - Shacharit, Mincha or Maariv - he would remain standing. And those of Titkabel and of returning the Torah Scroll [to the Ark], he would answer [PRESUMABLY Amein Yehei Shmei Rabbah] and afterwards he would sit.' Mishnah Berurah in Note 8 adds Hallel to the latter list of non-post-Amidah Kaddishes, then opines, "It is proper to concern oneself with the words of the more stringent opinions. And we may derive this a priori from Eglon, King of Moab who was a Gentile, yet he rose on his own from his throne due to G-d's words. Surely we, His Nation [should rise upon hearing holy words]!"
There are some who believe responding "amen" and "y'hay shmay rahbah..." in response to the Kaddish is one of the most essential aspects of prayer. In honor of this it would be logical to stand. (Of course, one could say the same for Shema Yisroel, the basic declaration of faith, during which we don't always stand!) Art Scroll's Mesorah Series "Kaddish" states that responding to Kaddish "takes precedence over an opportunity to respond to any other prayer, even Kedushah and Borchu" citing Orach Chaim (56:1) among others.It also says taht some congregations have a custom to say certain passages from Numbers and Psalms before recitation of the Kaddish. It does not appear to discuss the need for the congregation to stand. Neither the Art Scroll RCA or Koren siddurim seem to have directions to stand.
The Magen avraham and other poskim write that the maharil would only stand if he was standing up to the point of kadish,but if he was sitting he would stay that way, and so was the minhag of the arizal. It is good to note that the poskim are more machmir by kadish tiskabel since it is a essential part of the davening. The mishna berurah writes that according to everyone it is permissible to sit after the frase leala.