Standing for Kaddish and Barchu:
Some Poskim rule it is not necessary [for the listeners] to stand during the recital of Kaddish and Barchu. Other Poskim rule one is required to stand upon answering Kaddish or any Davar Shebekedusha [i.e. a matter that requires a Minyan]. Practically, it is proper to suspect for their words. [The Ashkenazi custom is to stand for Kaddish. The custom of Sefaradi Jewry is not to stand. Regarding the Chabad custom: The Rebbe was witnessed to stand for most of the Davening, including the Kaddeishim, and in those times that he was sitting, the Rebbe would stand when the Chazan reached Yihei Shmei Raba.]
If one was already standing, may one sit down? Even according to the lenient opinion, in the event that one was already standing prior to the recital of Kaddish, such as after Hallel [or after Yishtabach, Tachanun, Aleinu, and the like], it is good/proper to remain standing, and not sit down, until after answering Amen Yihei Shmei Raba. According to the stringent opinion, one must remain standing.
The Chazan: The Chazan, [or other person reciting Kaddish, according to all opinions] is required to stand for the recital of Kaddish, Barchu, and any other Davar Shebikidusha.
It is proper for one to stand during Kaddish, Barchu and other Davar Shebekidusha. This especially applies in a case that one was already standing, in which case he is not to sit down during Kaddish, until at least after Amen Yihei Shmei Raba. [However, the custom is to remain standing until the end of Kaddish-see Q&A!]
Until where in the Kaddish is one to remain standing?
Some Poskim rule one is to stand until the conclusion of Amen Yihei Shimei Raba. Other Poskim rule one is to stand until after the Amen of Deamiran Bealma. [Practically however, the widespread custom is to stand throughout the entire Kaddish, and so should be followed. When the Rebbe would stand for Kaddish, he would remain standing until the conclusion of the entire Kaddish.]
 Admur 56/5; See Kaf Hachaim 56/20-22; Piskeiy Teshuvos 56/4; Admur ibid concludes in parentheses: (See Chapter 146/4 for a similar discussion regarding if one must stand for Kerias Hatorah.)
 1st opinion in Admur ibid, brought also in 53/1; Maharil Tefila 3, brought in Darkei Moshe 56/5; Arizal Shaar Hakavanos Kaddish; Peri Eitz Chaim Shaar Hakadeishim 6; brought in M”A 56/4 and Taz 53/1 and 56/2
 2nd opinion in Admur ibid; Rama 56/1; Hagahos Minhagim in name of Yerushalmi [Arizal ibid claims is misprint in Yerushalmi]; Shiltei Giborim on Mordechai Birchas Hashachar 5; Reishis Chochma Shaar Hayirah 15/59
 The reason: One can learn this as a Kal Vachomer from Eglon. The verse [Shoftim 3/20] states that Eglon, the king of Moav, stood up on his own from his throne upon hearing the word of Hashem. [Now, if a gentile got up in honor of Hashem then] certainly then by us, Hashem’s nation, [we should stand up in His honor]. [Admur ibid; M”A ibid]
 Admur ibid; M”A ibid “one is not to be lenient”; M”B 56/8; Aruch Hashulchan 56/9 that the custom is stand only for the necessary Kaddeishim that must be said during the prayer
Other opinions: Many Poskim conclude like the ruling of the lenient opinion and custom of Arizal. [Kneses Hagedola 55/1; Yad Ahron; Shalmei Tzibur p. 81; Kesher Gudal 8/14; Siddur Beis Oved; Chesed Lealafim 56/7; Kaf Hachaim [Falagi] 13/7; Yifei Laleiv 56/3; Ben Ish Chaiy Vayechi 8; Kaf Hachaim 56/20 that so is custom]
 Sefardi Poskim brought under the lenient opinion
 Pinat Halacha of Sichat Hashavua, by Harav Yosef Simcha Ginzberg
 Admur ibid; M”A ibid; Darkei Moshe 56 that so was custom of Maharil; Shaar Hakavanos ibid that so was custom of Arizal; Chesed Lealafim ibid; Ben Ish Chaiy ibid; Many Achronim brought in footnote under lenient opinion
 Admur ibid; M”A ibid does not mention that it is merely “proper” to do so, but simply states that so was the custom of the Maharil; Many of the other Poskim ibid write that one must remain standing and the Kaf Hachaim 56/22 writes it is Mitzvah to remind one who is coming to sit down in middle of Kaddish that he may not sit.
 Admur 53/1; Bach 53; Taz 53/1; P”M 53 M”Z 1
 M”B 56/7
 Admur ibid regarding one who was standing when Kaddish began; Maharil brought in M”B ibid
 Elya Raba, brought in M”B ibid
 Halichos Shlomo 6; Piskeiy Teshuvos 56/4