Half Kaddish:

Half Kaddish:[1] After the completion of the reading, Half Kaddish is recited. The custom is to cover the Sefer Torah while it is recited.

When:[2] On Shabbos and Yom Tov, the Kaddish is recited prior to Maftir. On all other days of reading, the Kaddish is said after the completion of all the Aliyos [i.e. Monday, Thursday, Mincha of Shabbos, fast day, Chol Hamoed].

Hagomel:[3] Those who desire to recite the blessing of Hagomel after the last Aliya before Kaddish are not to do so until after the half Kaddish is recited.

An Avel:[4]

The half Kaddish recited after Kerias Hatorah is designated to be recited by Aveilim.[5] This applies even if he did not receive an Aliyah and is not the Baal Korei.[6] Others[7] however rule that it is to be recited only by the Baal Korei. Practically, the Chabad custom is for an Avel, and Baal Yartzite, to endeavor to recite the half Kaddish after the Torah reading.[8] [This applies even on Shabbos and Yom Tov.[9]] If however an Avel is not available, then it is to be recited by the Baal Korei.[10] Likewise, in a case of dispute, or when Davening in a Shul with a different custom, the Kaddish is to be recited by the Baal Korei.[11]

13. If two or more Torah scrolls are removed:^[13]

Whenever two scrolls are removed, the weekly Torah portion is read from the first scroll until Shevii. The second scroll is then placed on the Bima which is followed by half Kaddish.^[14] Hagba is then done to the first scroll. [The Mi Shebeirach for the ill is recited after Hagba.] One may not open the second scroll until the first scroll is rolled up and placed in its Meil.[15] For Maftir the portion of Shekalim is read from the second scroll. One then performs Hagba to the second scroll and reads the Haftorah for Shekalim. [On Shabbos Mevarchim the Chazan for Musaf takes the scroll of Maftir to hold.[16] When the scrolls are being returned to the Aron, the scroll of Maftir is taken first.[17]] 

When three Sifreiy Torah are removed: The weekly Sedra [Parsha of the week] is read regularly from the first scroll [until the 6^th Aliyah. By the 6^th Aliyah the remainder of the Parsha is read from Shishi until the end of the Parsha.^[18] One then places the second scroll on the Bima for the seventh Aliyah and does Hagba to the first Sefer Torah.[19] One may not open the second scroll until the first scroll is rolled up and placed in its Meil.[20]] The portion of Rosh Chodesh is then read from the second scroll. [One then places the third scroll on the Bima together with the second scroll and says half Kaddish.^[21] Hagba is then done to the second scroll. The Mi Shebeirach for the ill is recited after Hagba.] This is followed by the reading Maftir.

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[1] Ketzos Hashulchan 25:13

[2] Admur 282:12-5

[3] Shaareiy Efraim 19:69; Sefer Haminhagim p. 14; Toras Menachem 36:134; Shulchan Menachem 1:262

Other opinions:

[4] See Piskeiy Teshuvos 147/9; Pnei Baruch 34/14; Nitei Gavriel Vol. 2 chapter 45

[5] Sdei Chemed Kaddish 143 in name of Agudas Eizov 7, based on based on Rashbatz 3/171; Mateh Efraim Kaddish 3/1; Alef Hamagen Kaddish 3/3; Shaareiy Efraim 10/9 regarding if the Avel received Maftir or Shelishi; Gesher Hachaim 8-2; Pnei Baruch ibid;

[6] Alef Hamagen ibid, however see Shaareiy Efraim in previous and next footnote

[7] Maharam Brisk 1/118; Mishmeres Shalom Kudinov on Midrash Pinchas 35; Sefer Hachaim 139/1; See Shaareiy Efraim ibid and Ketzos Hashulchan 84 footnote 3 and Torah Leshma 422 that imply it only belongs to the Avel if he says Maftir; See Halichos Shlomo 12/27 in name of Rav SZ”A and other Poskim in Nitei Gavriel ibid footnote 1 that this Kaddish belongs to the souls of the deceased that don’t have a person saying Kaddish for them and hence should not be said by an Avel

[8] Sefer Haminhagim p. 32 [English];

[9] Shaareiy Efraim 10/9 and Ketzos Hashulchan 84 footnote 3 regarding if the Avel received Maftir or Shelishi; However see Piskeiy Teshuvos ibid footnote 35 for Poskim who write the Avel is not to say it on Shabbos and Yom Tov

[10] Shevet Halevi 8/163; see Piskeiy Teshuvos ibid

[11] Nitei Gavriel 45/3 footnote 6 in name of Poskim

[12] Nitei Gavriel 45/3 footnote 7 in name of Agudas Eizov and Tashbatz ibid

[13] Mishneh Berurah 685:9 and 12

[14] Rama 147:8; See 282:12 for the reason why Kaddish is recited between the five Aliyos and Maftir.

[15] Michaber 147:8; Kol Bo; Elya Raba 147:9; Kaf Hachaim 147:42; Ketzos Hashulchan 84:2

[16] Shaar Efraim 10:41; Piskeiy Teshuvah 150:4

Bedieved: If the Chazan accidently took the first scroll, it is not to be switched for the second scroll due to worry of Pegam. [Shaareiy Efraim ibid]

[17] See Shaar Efraim ibid

[18] Mishneh Berurah 685:4

[19] Hagba is only done after the second scroll is placed on the Bima. [Rama 147:8] Some [Rav M. Harlig] say that the Rebbe instructed that Torah is to be placed on the right side of the first Torah. Others say it is to be placed on the left side. [See Piskeiy Teshuvos 147 footnote 36; Hiskashrus 1078 footnote 5]

[20] Michaber 147:8; Kol Bo; Elya Raba 147:9; Kaf Hachaim 147:42

[21] Mishneh Berurah 685:5; See Levush and Kaf Hachaim 684:19 for the reason behind this; The Sefaradi custom is to say two Kaddeishim, one after the second scroll, and one after the third scroll [Kaf Hachaim ibid]