According to sources from the shulchan aruch harav's website:
Is Christianity considered Avoda Zarah?
Christianity is considered Avoda Zara for a Jew, and carries all Avoda Zara related prohibitions. However, some Poskim rule that for gentiles, Christianity is considered Shituf and is not prohibited due to Avoda Zara. Other Poskim however rule that Christianity is considered like idolatry even for gentiles.
 Rambam Machalos Assuros 11:7; Avoda Zara 9:4; Pirush Hamishnayos Avoda Zara 1:3; Teshuvas Harambam 448; Rama Y.D. 148:12 [in uncensored editions] lists Xmas and New Years as Holidays of idolatry; Likkutei Sichos 37 p. 198; Rebbe in handwritten editing remarks to a letter “Christianity is Avoda Zara, is in contrast to the seven Nohadite laws, as opposed to Islam. However, the Christians of today are simply “Maaseh Avoseihem Beyadeihem”.
The reason: As they believe that Yoshka is one of the three parts of Hashem and they worship him. [In truth however, there are different sects of Christianity with different belief systems. See Haemuna Vehadeios of Rasag 2:7 that there are four groups of Christians and not all are idol worshipers; See here https://www.thoughtco.com/faith-groups-that-reject-trinity-doctrine-700367]
 Rama O.C. 156:1 [Omitted in Admur 156]; Darkei Moshe O.C. 156; Y.D. 151; Shach Y.D 151:7; Tosafus Sanhedrin 63b and Bechoros 2b; Ran Sanhedrin 63b, end of first Perek in Avoda Zara; Meiri Bava Kama 113b; Rabbeinu Yerucham Nesiv 17:5; Beir Hagoleh C. M. 425 Shin; Aruch Hashulchan 156:4; Reb Yeshaya Berlin, brought in Mishnas Chachomim Yesodei Hatorah Lav Alef and Pischeiy Teshuvah 147:2; See Sefer Hamamarim Rebbe Maharash 5637 “Mi Kamocha”; Melukat 1 p. 323 Mamar Mayim Rabim 5717; Melukat 3 Nissan p. 128 Mamar “Beyom Ashtei Asar” 5731; Toras Menachem 5743 3 p. 1386; See Pischeiy Teshuvah Y.D. 157:2 that so is clearly implied from Rama ibid; See Sdei Chemed Peas Hasadeh Kelalim 3:6
 The reason: The definition of Nohadite idolatry according to this opinion: According to this opinion, the form of idolatry prohibited for Nohadites is the belief that Hashem has completely left the earth, and plays no role in it. That it has been left completely to the authority of a deity to whom they pray and worship. In other words, they believe in G-d’s existence, but simply call him “Eloka Dielokaya/The G-d of G-d’s/.” This is prohibited for even a gentile to believe, and he must believe that G-d has direct influence on the world. It goes without saying that denial of G-d’s existence at all, and believe in a foreign deity, is defined as idolatry. However, Shituf, which is permitted for a gentile according to this opinion, believes Hashem interacts with the world, although has given authority to other deities or powers to also have some control in the world, and He is thus not the sole ruler. While Jews are prohibited from believing this due to idolatry, gentiles are not. [See Sefer Hamamarim Melukat 1:323, Mamar Mayim Rabim 5717 Melukat 3 ibid footnote 20]
 Noda Beyehuda Tinyana Y.D. 148 [Says that Shituf is Avoda Zara even for gentiles, and the Rama and Ran never intended to say that Shituf is permitted for them, and thus the common statement of people that based on the Rama there is no Issur of Shituf for gentiles, is incorrect; However, see Pischeiy Teshuvah ibid that in truth one must say the Rama holds Shituf is permitted for gentiles, as the Rama Y.D. 151:1 permits selling them items, as explains Shach 151:7]; Mahara Sasson; Vishev Hakohen 38; Meil Tzedaka 22; Shaar Efraim 24; P”M 156; Y.D. 65 S”D end of chapter that is nevertheless Rabbinically prohibited; Binyan Tziyon 1:16; Mishnas Chachomim Yesodei Hatorah Lav Alef; Pischeiy Teshuvah Y.D. 147:2 [concludes that Shituf is forbidden also for gentiles]; Rebbe in handwritten editing remarks to a letter “Christianity is Avoda Zara, is in contrast to the seven Nohadite laws, as opposed to Islam. However, the Christians of today are simply “Maaseh Avoseihem Beyadeihem”.
May one say the name Jesus or Yeshu?
From the letter of the law, it is permitted to recite the name Jesus or Yeshu. It is likewise permitted to write these names, as we find Gedolei Yisrael who wrote these names in their Sefarim. Nevertheless, despite the letter of the law, the custom of all Jewry dating back many generations is to avoid saying these names and rather the term Oso IshÃ‚Â or YoshkaÃ‚Â or Yoshke PandreÃ‚Â is used in its stead. One is not Heaven forefend to break this custom.
May one say the word Christ?
One is not to use this term as it connotes a Messiah and savior, and according to some even a deity, and so is the custom of all Jewry to not say this term.
May one say the name Chris-mass?
No, and so is the custom. One is rather to use a epithet [i.e. nickname] such as Kratzmacht; Nittel, and the like. Seemingly, however, the term X-mass is not to be used, as the X is short for Ch***, and is used also by Christians as a formal name of the holiday.
May one say the name Mary?
From the letter of the law, it is permitted to do so, although G-d fearing Jews avoid saying this name [when in reference to the mother of Yoshka].
May one say the name Muhammad or other followers of Islam?
Yes, as the religion of Islam is a true monotheistic religion which carries no aspect of idolatry.
 The reason: As it is permitted to recite names of people who do not connote a deity and were later turned into a deity. [Yireim brought in Hagahos Maimanis Avodas Kochavim 5/3; Chavos Yair Teshuvah 1 Hasaga 11-12; Biur HagrÃƒÂ¢Ã¢â€šÂ¬Ã‚Âa 147/3; Teshuvos Rav Ezriel Hildsheimer 180; See Mishneh Halachos 9/169]
 See Biur HagrÃƒÂ¢Ã¢â€šÂ¬Ã‚Âa ibid; This name is mentioned in various places in the Gemara and Rambam end of Mishneh Torah-uncensored version; See Sanhedrin 43; 67; 105; 107; Avoda Zara 27; Yerushalmi Brachos 5/1; Rambam Melachim However see Teshuvos Rav Ezriel Hildsheimer 180 that no proof can be brought from here that these names may be recited as there is no prohibition to write the names, and rather the prohibition is simply to say them.
 Teshuvos Rav Ezriel Hildsheimer 180; See Mishneh Halachos 9/169
 Teshuvos Rav Ezriel Hildsheimer 180; See Mishneh Halachos 9/169
 Although we find Sefarim that write this term as well, nevertheless one cannot learn from them that it is permitted to say the names, as writing is more lenient than saying, as well as that it is permitted to write the names for learning purposes. [ibid]
 See here http://www.thefreedictionary.com/Xmass
 Chavos Yair Teshuvah 1 Hasaga 11-12
 The reason: In addition to all the reasons of allowance mentioned above regarding Yoshka, this woman is not worshiped or considered a G-d by even the Christians, and hence it has no relation to idolatry. [ibid