Trimming the mustache during the three weeks:
One may trim any mustache hair that interferes with him eating. [This applies even during the week of Tisha B’av. However, some Poskim rule that during the week that Tisha B’av falls in, it is forbidden to trim any mustache hair, even if it interferes with food.]
May one trim his mustache?
Many Poskim and Kaballists rule that one is obligated to cut the mustache hairs that interfere with ones food. Nevertheless, there are communities that are accustomed not to trim any facial hair, including the mustache hairs that interfere with food. The Rebbe answered that with regards to cutting the mustache there are sides to say it is permitted and sides to say it is forbidden and one should therefore ask Rabbanei Anash.
Washing hands: One who trims his mustache is not required to wash his hands afterwards.
 Michaber 551:13; Tur in name of Ramban
A Mohel: A Mohel may trim any hair that interferes with Metzitza. [Kaf Hachaim 551:172 in name of Ikarei Hadaat 27:11]
 Michaber ibid simply writes hair of the beard that interferes with eating. However we are accustomed to forbid cutting any hair of the beard other than the mustache.
 So is clearly understood from Michaber ibid who in the previous Halacha regarding haircuts only prohibits it on the week of Tisha B’av, and on this he states that trimming the mustache is permitted; Lechem Hapanim 122:3 in opinion of Michaber; Kneses Hagedola 551:13; Mateh Yehuda; Mamar Mordechai 551:12; Kaf Hachaim 551:170
The reason: As the week of Tisha B’av is similar to Shloshim, and not to Shiva. [Poskim ibid] And during Shloshim it is permitted to trim the mustache. [Michaber Y.D. 390:1; Maharitz Geios brought in Tur 390]
 Kitzur SHU”A 122:4; Ateres Zikeinim 551:13 that according to the Rambam and Raavad, brought in Shach 390:1 and Beir Hagoleh 390, and so rules Beis Yosef 390, that even within Shloshim it is forbidden to trim the mustache
 Moed Katan 18a; 531/8 regarding Chol Hamoed; 551/13 regarding three weeks; Yoreh Deah 390/1 regarding Avel; Shulchan Gavoa Yoreh Deah 181/17, brought in Zivcheiy Tzedek 181; Darkei Teshuvah 181/22; Yosef Ometz Dinei Peios; Yifei Laleiv 3/181; Ben Ish Chaiy in Torah Leshma 391; See Sefer “Hadras Pnei Zakein” Chapter 11 p. 508 and 773
 Zohar Idra Raba 134a; Custom of Arizal mentioned in Shaar Hamitzvos, Taamei Hamitzvos, Likkutei Torah Kedoshim and many other Sefarim; Emek Hamelech Shaar Reisha 61; See Sefer “Hadras Pnei Zakein” Chapter 11 p. 508
 The reason: 1) As this causes one to stumble in eating meat and milk due to remnant that remain by his mustache hairs. [Shulchan Gavoa] 2) As this is the ways of the gentiles, and makes one look like a thug. [ibid]; 3) As this prevents one from articulating his prayers properly. [Emek Hamelech ibid; Yifei Lalev ibid] 3) As this prevents one from eating properly which causes a blemish above. [Ben Ish Chaiy ibid] 4) As the hairs that interfere are considered putrid and they blemish the food. [Yaavetz]
 See Daas Moshe p. 242; Sefer “Hadras Pnei Zakein” p. 774 and the testimony of Rav Groner in next footnote
 Igros Kodesh 19/422 [printed in Shulchan Menachem 4/36 p. 133]
Rabbi Ashkenazi’s reply to my query: The answer I [i.e. the author] received to this question from Rav Mordechai Ashkenazi, the Rav of Kfar Chabad, is that one may trim the mustache hairs that are in the way of food. [The question was asked as follows: “Is there any Hiddur involved not to cut the mustache even if it is in the way of food?”]
Rabbi Leibal Groners reply to my query: “The Rebbe never trimmed his mustache, neither did any of the Elder Chassidim or even young ones. there were individuals that did but a very small number.”
 Oar Letziyon 2/44-5; Halichos Shlomo 2/10