Most Chassidic Groups:1
On the yahrtzeit of a rabbi with connection to the group (either a previous rebbe, or someone with substantial influence on the group or that town's inhabitants).
When their rebbe does, e.g. when his child gets married or is circumcised.
On days when special joyous events happened to their past rebbes, e.g. they were freed from ...
The Ramma in siman 131 siff 2 brings from the Beis Yosef who quotes a Rokeach --
The Rokeach also writes in siman 324 not to 'fall' unless in front of a seffer torah. And a siman for this is found in the war against Ay (Yehoshua 7, 6) 'and he fel lon his face in front of the aron'."
The Beis Yosef concludes by saying if it's accepted (kabbala), he'll ...
The Rema in Shulchan Aruch (131:1) writes that at Shacharit (or Mincha;see Mishne Berurah 7) when one wears tefillin, to lean on the right in deference to the tefillin and when not wearing tefillin (e.g. during Mincha or even Shacharit if there is no tefillin on his left arm) he leans on the left side.
Per Rabbi Avraham Yosef Shlita if someone started saying Kaddish in error and people already answered the first Amen it should be completed until Chatzi Kaddish.
Piskei Teshuvos 131:4 notes 14 & 15 says as follows.
Shaalos U'Teshuvos Lehoros Noson Volume 6 Siman 7 says that when Kaddish was said in error, Tachanun should be recited immediately and then ...
The source of the Rambam's ruling is Taanis 14b:
אמר ר' אלעזר אין אדם חשוב רשאי ליפול על פניו אלא אם כן נענה כיהושע בן נון שנאמר
(יהושע ז, י) ויאמר ה' אל יהושע קום לך למה זה אתה נופל על פניך
R. Elazar said: A important person should not fall upon his face unless he is confident that he will be answered like Yehoshua, as it is written, And the Lord ...
Rivivos Efraim 8 page 400 says that one should say Tachanun at a Bar Mitzva and also has a letter from Rabbi Nosson Gestetner which says that Tachanun should be said at a Bar Mitzva as it is not listed in Orach Chaim 131:4-6 as one of the times it should not be said.
See also Minchas Yitzchak 8:11 where he mentions that the Minchas Elazar did not say ...
The Rambam is referencing a Gemara (Megillah 22b) which says:
אמר רבי אלעזר, אין אדם חשוב רשאי ליפול על פניו אלא אם כן נענה כיהושע בן נון, דכתיב +יהושע ז:י+ ויאמר ה' אל יהושע קם לך [למה זה אתה נפל על פניך]
Rabbi Elazar said: An important person is not allowed to fall on his face unless he is [sure to be] answered like Yehoshua bin Nun, as it ...
Rav Ovadia Yosef (Yabia Omer OC 4:13) discusses this topic and concludes in favor of not saying tachanun at a siyum in order to emphasize the aspect of happiness when learning Torah and thereby reward those who do learn Torah / entice others to learn Torah too.
Questions 1 and 2:
O Ch 131 (4) MB  says that those davenning in the courtyard which does not have a separate ark or amud are included in the same law as the main synagogue with respect to not saying tachanun when there is a choson or a bris.
(The “Dirshu” MB in note 21 on O Ch 131 (4) MB  writes that the Ladies Area “Ezras Noshim” is considered ...
Taamei HaMinhagim, entry 128, says that we do not say Tachanun on the eve of Shabbat or Yom Tov (and other eves), as a reminder to pray the special prayers in that evening's prayers, or at least as a reminder that the special day that starts that evening.
Those who do say Tachanun probably do so because 14 Iyar is not listed as a day where Tachanun was customarily omitted by the Tur, anyone quoted in the Beit Yosef or Bach, by the Shulchan Aruch and its Mapah, the Levush, the Eliya Rabba, the Taz, the Magen Avraham, the Beiur haGra, the Chayei Adam, the Kitzur Shulchan Aruch, the Mishna Berura and others. ...
Nissan starts without Tachnun because it is the time when the Mishkan was inaugurated - that is the first 12 days. Then the 13th is the Isru Chag of those 12 days (The Tzemach Tzedek quoting the Maharil), and then there is no Tachnun on the 14th because the Korban Pesach was brought, and then Pesach. So there is no opportunity to say Tachanun until the ...
[Collected from the writings of Ha-Rav Aviner]
Tachanun on Erev Yom Ha-Atzmaut
It is proper to recite Tachanun at Minchah of Erev Yom Ha-Atzamaut, as the Chief Rabbinate of Israel has decreed this day "Remembrance Day for the Fallen Soldiers of Tzahal," which is a day of mourning.
Iturei Cohanim #97, Shut She'eilat Shlomo 3:147, Sichot Ha-Rav Tzvi ...
The Siddur of the Alter Rebbe writes:
"Minhag Sefarad: Any day on which Tachanun isn't said, Lamnatzeyach and Tefilla LeDovid aren't said either. For example, the whole month of Nissan, Peisach Sheini ... or any day on which their is a Bris in the Shul, or a Chosson for the [seven] days of partying" (emphasis mine).
These aren't calendrical days.
I'm not sure about Chabad, but Sepharadim either include all of the following or omit all of the following. They never say only some of them:
Two other parts of the service have the exact same conditions on when they are said as the above three parts, with the added factor that they are only said on certain days of the week.
Siach Tefila question 137 & 138 indicates as follows. If one is in the midst of the actual Tachanun (the part where you place your head on your arm) then he should finish the Tachanun, however if one is in any other part then he should stop and say Vayehi Bin'soa.
The Aruch Hashulchan (131:7) and Mishnah Berurah (131:3) both say that one should cover his face with a cloth and not merely one's hand, because a person's own flesh can't be considering a covering for itself.
If one is wearing short sleeves (assuming that one is allowed to pray in short sleeves...) than it appears that covering one's face with the arm ...
In SA OC 607 the Magen Avraham brings two reasons for this custom:
1) The Ari Z"L says that one should pound on their chest
2) A Medrish Kohelet that says "We bang on our hearts to show that that it (the heart) is what lead us astray"
Now I think both sources are expressing the same idea BUT in this case perhaps we can say that -
According to the ARI Z"L ...
Qitzur Shulhhan Arukh - Yalqut Yosef (Orahh Hayim 131:36) states:
אלו הימים שאין אומרים בהם וידוי ונפילת אפים לא בשחרית ולא במנחה: חודש ניסן, פסח שני, ל''ג בעומר, מראש חודש סיון עד י''ב בו ועד בכלל, תשעה באב, ט''ו באב, ערב ראש השנה וערב יום הכפורים, מי''א תשרי עד ב' חשון. שמונת ימי החנוכה, ט''ו בשבט, י''ד וט''ו אדר א' ואדר ב'. וראשי חודשים.
ואפילו בימים שאין אומרים תחנון אומרים למנצח מלבד בראש חודש וחנוכה ופורים וע"פ וערב יום כפור וט"ו באב [מנהגים וע"ל סי' תקנ"ט]:
Even on days where tachanun is not said we say Lamnatzeiach except Rosh Chodesh, Chanukah, Purim, Erev Pesach, Erev Yom Kippur, and Tu B'Av.
Rema to OC 131:1.
The Mishnah Berura adds (131:35):
והנה כל אלו הסעיפים הוא לענין ...
See here where they suggest:
1 they often daven until after shkiya, and according to many opinions
tachanun may not be said after shkiya, so a blanket rule was
instituted so as never to come to saying after shkiya, which in some
kabbalistic sources danger is associated with this practice.
2 tachanun by mincha requires intense concentration, ...
From R' Dov Lior's answer at Yeshiva.org.il:
הסיבה לכך היא, שכל עניין "פסח שני" מתחיל מהבוקר ואין שום עניין בלילה. את קרבן פסח שני עשו בבוקר, ובלילה לא עושים שום דבר (בניגוד לשאר חגים). ולכן כל העדות צריכות לומר תחנון.
The reason for this is that the whole concept Pesach sheini starts from the morning and there is nothing related to the evening. ...
It seems from this story that if an individual doesn't say Tachanun for most of the month for a private reason (with exception of Nissan where everyone doesn't say Tachanun for most of the month and therefore the whole month) one still does Tachanun at any given opportunity when one can
Bava Metzia 59b
אימא שלום דביתהו דר"א אחתיה דר"ג הואי מההוא מעשה ...
Igros Moshe Orach Chaim 3:89 - last paragraph says that if ones Nusach is not to say Vidui & 13 Midos, and is in a Shul that does say Vidui & 13 Midos, he should definitely say the Vidui & 13 Midos. He says one has to follow the Minhag of the Shul he is Davening in, in any situation where it is noticeable.
Rabbi David Lau says that one who Daven'...