General laws applicable on fast days:
May one brush ones teeth on a fast day?
If not doing so will cause one pain or discomfort it is allowed.
May one use mouthwash if not doing so will cause pain and discomfort?
Yes it may be used in such a case.
May one bathe on the 10th of Teves?
Some are stringent to avoid bathing on all the four fasts.
May one listen to music and dance on the 10th of Teves?
Some are accustomed to be stringent in this matter.
 Michaber 567/3; Sefer Haminhagim p. 4 [English]; Ketzos Hashulchan 2 footnote 25 in name of Ateres Zikeinim
 M”B 567/11
 So rules Minchas Yitzchak based on 567/3 M”B 11 which allows to rinse ones mouth in a case of pain
 Kinyan Torah 2/49; See Mateh Efraim 602/3; Piskeiy Teshuvos 567 footnote 3
 Based on Bach brought in M”B 550/6 that a Baal Nefesh is to be stringent to follow all the oppressions of Tishe Beav on all the other fasts as well.
 See Biur Halacha 552 “Merosh Chodesh”; Kitzur SHU”A 122/1 “On the 17th of Tamuz and the tenth of Teves it is forbidden”