Joining a Minyan from a different area:
Answering Kaddish and Kedusha from a different area than the Minyan: Once there is Minyan of ten Jews in one area that are saying Kaddish or Kedusha, anyone who hears their voices can answer with them. This applies even if one is in a completely different house and there are many houses that intervene between them. It is considered as if the person hearing the voice is together with them in the Minyan.
Being Yotzei a blessing from a different area: The same applies regarding being Yotzei [Shemoneh Esrei during Chazaras Hashatz] that only the Chazan and the congregation must be in the same room, however once they are in the same room, anyone who can hear the voice of the Chazan can be Yotzei even though he is not with them. The same applies regarding being Yotzei Kiddush, Havdala, and Megillah from a different area [and being included in the blessing of Birchas Kohanim, and doing Nefilas Apayim with the congregation].
If there are feces or idolatry in-between: Some Poskim rule that one may only answer for [Amen] Kaddish or Kedusha [or be Yotzei a blessing] from a different area than the Minyan if there is no feces, or idolatry which is considered like feces regarding this matter, found in-between the two areas. [Other Poskim however argue that feces and idolatry do not intervene.] Practically, [although the main ruling is like the lenient opinion] one is [nevertheless] to suspect for the stringent opinion. [However in a time of need, some Poskim rule one may choose to follow the lenient opinion. Others rule one is always to be stringent.]
Once there is a Minyan in the room, any other person can join the Minyan and answer Amen and Kedusha even if he is in a different room or building, from which the voice of the Chazan can be heard. One may likewise be Yotzei a Mitzvah, such as Kiddush, Havdala, Megillah and the like from a different room. If however there are feces or idolatry in-between the two areas then one is not to answer [Amen], or to Kaddish and Kedusha [or be Yotzei a Mitzvah].
Must one answer Kaddish and Kedusha of a Minyan that he is not participating in and is in a different area?
If one is not in the same room as the Minyan and hears Kaddish or Kedusha, some Poskim rule that although he is allowed to answer to the Minyan, nevertheless he is not obligated to do so. Other Poskim however rule that he is obligated to answer, even if he is in another room and is not participating in the Minyan. If however one is in the middle of a Mitzvah, such as Davening or learning Torah, he is not obligated to answer.
Q&A on intervening items
Do feces and idols intervene before the answerer if he is found within the same room as the Minyan?
Some Poskim rule it is not considered an interval, and thus if one is distanced four Amos from the feces and it is not within his sight, he may Daven and answer to the Minyan. Other Poskim however rule it is considered an interval, and those people who are in-between the feces and the Minyan cannot answer Amen, or to Kaddish and Kedusha. Practically, we rule like the former opinion.
Is a church defined as idolatry and thus intervenes before the answerer?
Yes. [Thus, people that are found behind a church are not to answer to a blessing or Minyan taking place in front of the church. This can occur during rallies in New York city that have thousands of people filling the streets, and some are positioned opposite a church.]
May one answer and be Yotzei from a different area if there is a potty or bathroom in-between the two areas?
May one answer and be Yotzei from a different area if there is urine in-between the two areas?
Yes, so long as one is distanced from it four Amos and it is not within his view.
May one answer and be Yotzei from a different area if there is a gentile in-between the two areas?
Some Poskim rule that one may not answer Amen, or to Kaddish or Kedusha if there is a gentile in-between the Minyan and the individual. This applies whether the gentile is male or female. Other Poskim rule that one may answer for Amen, Kaddish and Kedusha even if there is an idol worshiper in-between. Practically, according to Admur a gentile does not invalidate answering Amen even if he is an idol worshiper.
May one answer and be Yotzei from a different area if there is a Mumar or Apikores in-between the two areas?
Some Poskim suggest that perhaps one may not answer Amen, or to Kaddish or Kedusha if there is a Mumar Apikores, or one who is not Shomer Shabbos, in-between the Minyan and the individual, just as is the law by a gentile. Practically however, according to Admur a gentile does not invalidate answering Amen even if he is an idol worshiper, and certainly a Jew does not invalidate even if he is an Apikores.
May one answer and be Yotzei from a different area if there is a woman in-between?
Yes. This applies even if she is an Erva or is not dressed modestly, so long as she is not within his view.
 Admur 55/22; Michaber 55/20; Pesachim 85b
 Admur ibid; Michaber ibid; Rebbe Yehoshua Ben Levi in Pesachim 85b
Other opinions in Gemara: Tosafus [Pesachim ibid] learns that according to Rebbe Yehuda in Pesachim ibid [who argues on Yehoshua Ben Levi] one cannot answer unless he is in the same room. However according to Rashi’s explanation possibly Rebbe Yehuda agrees with Rebbe Yehoshua on this point.
 See Q&A regarding if this is voluntary or obligatory!
 The reason: As even a divider of metal cannot separate between a Jew and their father in Heaven. [Admur ibid and 128/37 and 131/3; Rebbe Yehoshua Ben Levi ibid]
 Admur 131/3
 Admur 55/22; Mishneh Rosh Hashana 27b as explained in Tosafus ibid
 Admur 273/10 and 213/1 regarding that one is Yotzei Kiddush and Havdala even if they are not in the same room as the person saying it
 Admur 213/1
 Michaber 690/14 that one can assume that the Chazan has in mind to fulfill the Mitzvah for all the listeners, including listeners that are behind the Shul and cannot be seen.
 Admur 128/37
 Admur 131/3
 Admur 55/22
 Opinion in Admur ibid; Michaber 55/20; Bahag in name of Rav Yehudaiy Gaon in name of Rav Acha; Mahariy Abuhav in name of Orchos Chaim in name of Rav Achaiy Gaon
 Halef Lecha Shlomo 40; See Piskeiy Teshuvos 55/30; Minchas Elazar 2/72
Other opinions: Some Poskim rule the above law only applies to answering for a Davar Shebekidusha and not towards Amen. [Ashel Avraham Butchach 55; See Poskim in Piskeiy Teshuvos 55/30 footnote 212]
 Machatzis Hashekel 273/13 regarding Kiddush; M”B 194/8 regarding Birchas Hamazon; Vetzaruch Iyun from Admur ibid who positioned this opinion between the Halacha of Kedusha/Kaddish and being Yotzei, hence implying they only argue regarding if one may answer, Vetzrauch Iyun.
Other opinions: Some Poskim rule the above law only applies to answering for a Davar Shebekidusha and not towards Amen. [See Poskim in Piskeiy Teshuvos 55/30 footnote 212]
 First opinion in Michaber and Admur ibid;
 Implication of Michaber who only brings as Yeish Omrim, and Admur ibid who does not write Vichein Ikkur and simply writes to suspect for their words; Levushei Serud 55/20; Maaseh Rav 47; M”B 55/62 that it is implied from Michaber and Rama that they mainly hold like the lenient opinion; See Shulchan Hatahor 55/8
Other opinions: Some Poskim rule the main opinion follows the stringent opinion. [Kaf Hachaim 55/95]
 Admur ibid; Levushei Serud ibid; Kaf Hachaim 55/95
Other opinions: Some Poskim rule one is only to be stringent regarding Barchu and other matters which mention Hashem’s name, and not regarding Kaddish and Kedusha. [Makor Chaim 55/20; M”B 55/62 in name of Chayeh Adam; Ashel Avraham Butchach]
 Levushei Serud ibid
 Kaf Hachaim 55/95
 Implication of wording of Admur and Michaber ibid who write “he is allowed to answer” and not “he is obligated to answer”; See Halichos Shlomo 9 footnote 8; Ishei Yisrael 24 footnote 2; Tzitz Eliezer 11/3; Rivivos Efraim 1/89; Piskeiy Teshuvos 55/29
 Kaneh Bosem 3/4; Ishei Yisrael ibid in name of Chazon Ish; See also Igros Kodesh 15/185 regarding answering Kaddish and Kedusha between th Sel Yad and Shel Rosh that the wording of “permitted” used by Admur in the Siddur implies that one is obligated to answer
 Toras Chaim Sofer 66/8 that one who is involved in a Mitzvah is exempt from a Mitzvah; Shevet Halevi 9/43; Tzitz Eliezer 11/4; Salmas Chaim 62; Kinyan Torah 4/9; Pischa Zuta 5; Teshuvos Vehanhagos 2/73-74; Yabia Omer 9/3; Piskeiy Teshuvos 51/11 and 55/29; Vetzaruch Iyun from Igros Kodesh ibid that one is obligated to answer in-between the Shel Yad and Shel Rosh even though one is in the midst of a Mitzvah.
 Implication of Rama 79/1 and Admur 79/3; Machazik Bracha 55/14; Beis David 55/32; Chesed Lealafim 55; Ben Ish Chaoy Vayechi 7; Toras Chaim Sofer 55/22; Kaf Hachaim 55/94
 M”A 79/3 learns based on the above ruling that even within the same room it is considered an interval; See also Devar Shmuel 189
 As implied from Admur ibid who completely omits the ruling of the M”A and writes as rules Rama ibid
 See Minchas Elazar 2/72
 Piskeiy Teshuvos 55/30
 Afikei Maginim Biurim 12, brought in Piskeiy Teshuvos ibid footnote 205
 Biur Halacha 55/20 “Veyeish Omrim”
 PT 55/30
 Old dialects of the Shulchan Aruch in Michaber write “gentile” instead of idols and the censorship changed the wording; Beis Yosef 55; Perisha 55/10; Elya Raba 55/18; P”M 55 A”A 15; Beis David 32; Machazik Bracha 55/15; Siddur Beis Oved; Ben Ish Chaiy Vayechi 7; Chesed Lealafim 55/6; Yifei Laleiv 5/1; Shaareiy Teshuvah 55/17; Toras Chaim Sofer 55/22; M”B 55/65 that either idols or idol worshipers intervene; Kaf Hachaim 55/92 in name of many Poskim and based on Zohar Mishpatim that all gentiles are invalid; M”B 55/65 that either idols or idol worshipers intervene
 Beis David ibid; Machazik Bracha 55/16; Shaareiy Teshuvah 55/17; Toras Chaim Sofer ibid
 M”A 55/15; Admur 55/22 based on the fact he writes that only idols are invalid due to being like feces; Devar Shmuel 189, brought in Beir Heiytiv 55/17; Minchas Elazar 2/72
 See Admur ibid
 Toras Chaim Sofer ibid; Piskeiy Teshuvos ibid
 See Admur ibid
 So is proven from the fact that it is omitted in Admur and all previous Poskim; Ashel Avraham Butchach 55 regrading if the male Erva is revealed in-between that it does not disqualify answering being one is not to add on the items listed by the Sages.
Other opinions: Some Poskim imply that if there is an Erva in-between the answerer and the person saying the blessing then it invalidates his ability to answer. [See Orach Neman 55/34; Piskeiy Teshuvos 55/30 footnote 203]
 Chasam Sofer Beitza 4b; Drashos Chasam Sofer 2/272; Michtav Sofer 2/4
 The reason: As it’s a Takanas Chachamim in memory of exile.
 Sichas Kodesh Simchas Torah 5749 7
 Hisvadyus 5791 1/207 that if Moshaich comes on Isru Chag then perhaps there should not be an Isru Chag; Sichas Shemini Atzeres 1939 “When Moshiach comes we will look back at the times we celebrated two days.”
 The reason: Several reasons can be suggested: 1) As we will sanctify the moon based on testimony and everyone will be able to know immediately when the month was set. 2) As the Kedusha of Eretz Yisrael will pass to all the Diaspora and there is hence no need for two days. [See Admur Basra 1/8]
 See Chayeh Levi 4/26; Piskeiy Teshuvos 285/9
 Kinyan Torah 6/12; See Betzeil Hachachmah 1/2-8
 Ketzos Hashulchan 72 footnote 3; Piskeiy Teshuvos 285/9
 Such as one traveled to Eretz Yisrael the week after a two day Shavuos or Pesach which coincided with Shabbos, in which case Eretz Yisrael is one Parsha ahead in its reading.
 Lehoros Nasan 3/13; Piskeiy Teshuvos 285/9
 Ikarei Hadaat 22/3; Ketzos Hashulchan 72 footnote 3; Piskeiy Teshuvos 285/9
 This can occur when the 2nd day of Pesach or Shavuos falls on Shabbos in the Diaspora and hence no Parsha is read, while in Eretz Yisrael the regular weekly Parsha was read. If one travels that week to the Diaspora he will be hearing the same Parsha that he heard the previous Shabbos in Eretz Yisrael.
 This ruling is evident from Admur 285/9 who does not require one to read Shnayim Mikra of the Yomim Tovim sections prior to each Yom Tov being that it was already read or will be read in its related Shabbos portion. Hence the same logistics apply here and there is no need to repeat Shnayim Mikra.