May one say the word Shalom in a bathroom or Mikveh?
Saying Shalom Aleichem: It is forbidden to greet a friend with the word “Shalom” [i.e. Shalom Aleichem], in the inner room of a bathhouse where everyone is undressed [i.e. where the showers and hot Mikveh is situated] being that Shalom is the name of G-d, and it is forbidden to mention the name of G-d in a filthy area. Likewise, it is forbidden to greet one’s friend with “Shalom” in filthy alleyways. [Accordingly, it goes without saying that it is forbidden to say the word Shalom in a bathroom, due to the above reason.] However, in the dressing room of the bathhouse where some people are dressed and some are undressed, it is permitted to say the word Shalom.
Calling one’s friend by the name Shalom: [It is disputed amongst the Poskim as to whether in a bathroom one may mention the names of people who are called Shalom.] Some Poskim rule that it is forbidden to call one’s friend by his name if his name is Shalom in the inner room of a bathhouse [or bathroom], and rather one is to call him by his name in a foreign language [i.e. if he has an English name]. Other Poskim, however, rule it is permitted to mention his name in a bathhouse [or bathroom] being that one has no intention to mention the concept of Shalom [i.e. name of G-d] and is rather simply calling his friend’s name. Practically, regarding a dispute in a Rabbinical prohibition one may follow the lenient opinion, and so is accustomed to be lenient and allow mentioning the names of people who are called Shalom in a bathroom or bathhouse. [However, a G-d fearing Jew is to be stringent and not say the name in full, but rather say Shalo without a Mem, or Shalon with a Nun. Or say the name Salom, with a Sin as opposed to a Shin. Or call him by a nickname, such as Shuli; Sheelem, Shulem.]
It is forbidden to say the word Shalom in a bathroom or shower room of a bathhouse or Mikveh, although one may be lenient to do so whenever his intent is not for the sake of saying the name of G-d, which is peace, such as when calling a friend whose name is Shalom. Nonetheless, even in such a case it is best to initially avoid the above and not say his name in full, and rather say Shalo, or Shalon, or Shuli, or Salom, etc.
May one say the word Shalom in a sentence when the context is to describe peace between two people or two nations?
Yes. [Nonetheless, based on the stringent opinion mentioned above, seemingly it is best even in such a case not to say the word in full.]
May one say the word “peace” in English?
 See Admur 84:1; 85:3; Michaber 84:1; Shabbos 10a; M”B 84:6; Kaf Hachaim 84:7; Piskeiy Teshuvos 84:2 and 5
 Admur 84:1; Michaber 84:1; Shabbos 10a
 The source that the name Shalom is the name of Hashem: As the verse [Shoftim 6:24] states “And they called Him G-d Shalom.” [Admur 84:1; M”B 84:6] See Rama Y.D. 276:13 in name of Tashbatz Katan 420 and Maharil, that some are particular not to write it in full, and rather write “שלו‘; See Nekudos Hakesef ibid that according to Tosafus Sotah 10a the name Shalom has Kedusha and may not be erased, and he concludes with a Tzaruh Iyun Lemaaseh; Radbaz 202, brought in Pischeiy Teshuvah 276:28; Mateh Efraim 581:9; Sdei Chemed Peas Hasadeh Alef Klal Kuf; M”B 84:6; Igros Moshe 4:40; Ginzei Hakodesh 7:8;
The meaning of the name Shalom: G-d is called Shalom when He expresses Himself to the worlds in a way of Peace, as is the case with all His names, that they represent a certain type of His expression. [Toras Menachem 19th Teves 1982]
 Admur ibid; M”A 84:2; M”B 84:6
 Admur 84:1 regarding the inner room of a bathhouse and filthy alleyways, and the same would apply even more so to a bathroom, which has a more severe status than a bathhouse, as explained in Admur 84:1; Piskeiy Teshuvos 84:4
 Admur ibid; Michaber ibid; Shabbos ibid
 Admur 84:1: “There are Poskim that prohibit calling one’s friend by his name if his name is Shalom, and rather one is to call him by his name in a foreign language. [Bach; M”A 84:2] Other Poskim rule it is permitted being that one has no intention to mention the concept of Shalom [peace] and is rather simply calling the friends name. [Taz 84:3] (Practically by Rabbinical matters one is to follow the lenient opinion and so is the custom.)” [parentheses in original]
 1st opinion in Admur ibid; M”A 84:2; Bach 84; Shiyurei Kneses Hagedoal 84:1; Peri Chadash 84 [forbids even in other language]; 1st opinion in M”B ibid
 2nd opinion in Admur ibid; Taz 84:3; Elya Raba 84:2; Halacha Berura 84:2; Soles Belula 84:2; P”M 85 A”A 3; Kisei Eliyahu 84:4; Birkeiy Yosef 85:9 based on Zekan Aaron 72; 2nd opinion in M”B ibid
 Admur ibid in parentheses; Ketzos Hashulchan 10:16 rules that practically the custom is to be lenient; [To note that in 85:3 Admur simply states that the prohibition of mentioning the name Shalom is when one intends on mentioning the concept of peace, hence implying that this is Admur’s final ruling.]; Tevuos Shur in Bechor Shur Sanhedrin 65a; M”B 84:6
 Birkeiy Yosef ibid; Chayeh Adam 3:36; Pesach Hadvir 85; Ruach Chaim 84:1; M”B 84:6; Kaf Hachaim ibid
 Chayeh Adam 3:36; Pesach Hadvir 85; Ruach Chaim 84:1; M”B 84:6; Kaf Hachaim ibid
 Pesach Hadvir ibid; Kaf Hachaim ibid
 Yifei Laleiv 84:1; Kaf Hachaim ibid
 See Beir Moshe 4:8 [writes that one may even ask Mah Shjelomcha!]; Piskeiy Teshuvos 84:5; See Radbaz 202, brought in Pischeiy Teshuvah 276:28, that only when Shalom is written in reference to Hashem, such as when blessing another Jew, is one to be careful, and not when the intend is for the word peace, such as that there is peace between two people
 See Admur 85:3; Bach 84; Olas Tamid 85:6; Ateres Zikeinim 84:3; Shach 179:11. [Kuntrus Achron 85:1] So rules also Kitzur SH”A 5:10; Chesed Lealafim 85:2
 Admur 84:2; M”A 84:2; Bach 84; Shiyurei Kneses Hagedoal 84:1; Peri Chadash 84 [forbids even in other language]; 1st opinion in M”B ibid