Can you wash clothes and get a haircut on a minor fast day?
Yes. Halachically we are even allowed to take showers, which may not be done on T’bav. There is an opinion that pious individuals should act on other fast days like on T’bav, by not washing themselves etc. This however is only for the things that are special to T’bav over other fasts like not bathing, but not to the other things that we don’t do customarily during the nine day.
See Biur Halacha 551-2 D:H M’rosh Chodesh that he writes from the Eliyahu Rabba that it is pious that one should be careful on 17 B’Tammuz, and 10 B’teves like the halachos of the nine days. However see Daas Torah 551-2 that what the Biur Halacha is bringing is based on a printing mistake, and also see Shoneh Halachos 551-38, that Horav C. Kanievsky also brings up this point. According to them, there wouldn’t even be an inyan to be machmir. Also see Chut Shani Shabbos 4- pg. 381, and Kara Olai Moed chap. 1 ftnt. 15 that even if it would be an inyan, it is only on things that are ossur min hadin from Rosh Chodesh, but not for things that we only because of a minhag from Rosh Chodesh, such as washing clothes and getting a haircut.