Early Mincha
There are a number of reasons why one should consider davening Mincha especially early (“mincha gedola”) when Asorah B’Teves falls on a Friday. On the most practical level, it allows the lengthy Mincha tefillah to be finished early in the day, allowing one to devote the remainder of the day to Shabbos preparations. Davening Mincha immediately prior to the arrival of Shabbos forces people into an earlier routine that they are not accustomed to, which is likely to cause distraction, disorderly Shabbos preparations, and widespread late arrival to shul.
Davening Mincha early in the day is also recommended as one will still be wearing one’s weekday clothes at that time. On the other hand, davening Mincha immediately prior to the arrival of Shabbos would force one to recite the tefillos of a fast day in one’s Shabbos clothes which is considered undesirable. (However, there are those of the opinion that one should always wear one’s Shabbos clothes for Mincha on Friday, even on Asorah B’Teves, regardless of the time of day one davens. See Nitei Gavriel 62:3).
It is also considered honorable and meritorious to distance the mournful tefillos of the fast day as far from the arrival of Shabbos as possible. (Devar Yehoshua 3:63, cited in Nitei Gavriel, Chanukah 63 note 4).
However, R’ Yaakov Zalzer pointed out to me, that in Eretz Yisroel things may be different. Last year when Asorah B’Teves fell out on erev Shabbos he wrote to me that: “You write that ideally one should daven early. However, there is a very big reason to daven Mincha Ketanah (at least for those of us in Eretz Yisroel), as that way there will be Birchas Kohanim!! (Also - specifically for those of us who follow the Mishnah Berurah and do not duchen by Minchah Gedolah, unlike minhag Bnei Brak based on the Chazon Ish).
Bathing/Showering and Haircuts on Asorah B’Teves That Falls on Erev Shabbos
The Rambam writes (Hilchos Ta’anis, 1:14): “Whenever a person is fasting, whether he is fasting because of an individual distress, a disturbing dream, or distress of a communal nature, he should not indulge in pleasures, act frivolously, or be happy and of good spirits. Instead, [his conduct] should be characterized by serious concern, [as if he were] in mourning, as [implied by Eichah 3:39]: “Over what should a living man be concerned? Each man over his sins.”
Following these directives, we find in the poskim various rulings regarding enjoyable pleasures on fast days:
On regular fast days, excluding Yom Kippur and the 9th of Av, only eating and drinking is forbidden. On Yom Kippur and the 9th of Av additional prohibitions apply (among them bathing, listening to music, laundering, wearing leather shoes). The poskim write that while those additional restrictions are not obligatory on regular fast days, a person of higher spiritual status should refrain from them.
The Mishnah Berurah writes (650:8) that even a person who does practice those additional restrictions may wash in cold water. The Aruch HaShulchan (550:3) explains that washing in cold water is not washing for pleasure but to remove discomfort, and as such, permitted on a regular fast day.
According to the Mishnah Berurah (Biur Halachah 551:2) on the fast days of the 17th of Tammuz and the 10th of Teves one should conduct himself as he does on the nine days leading up to the 9th of Av. This ruling is understood differently by the various contemporary poskim: According to the Tzitz Eliezer (7:49) this ruling is intended to restrict business dealings and starting new building projects, as well as marrying. The other restrictions that apply on those days before the 9th of Av do not apply (bathing, laundering and haircuts). However, Rabbi Seraya Devlitzky (Zeh HaShulchan volume 1, Orach Chaim 551) includes haircuts in this prohibition. Rabbi Chayim Palagi (Ruach Chaim 566:4) ruled that haircuts should not be taken on a fast day.
This year, when the 10th of Teves falls on Friday, the Mishnah Berurah writes (550:6) that in honor of Shabbos one can certainly bathe in hot water. Rabbi Chayim Palagi and Rabbi Seraya Devlitzky rule similarly regarding haircuts and laundering Shabbos clothes. (In general, though, one should refrain from doing laundry on Friday, if possible).
