Are the days during Sivan when we do not say Tachanun also asur in Taanis?
Yes. In principle, all days on which taanis is not recited are days on which it is forbidden to fast.
This applies even when a person has a yahrzeit, as the Chayei Adam (171:11) and the Kitzur Shulchan Aruch (221:7) rule. See also Mishnah Berurah 494:7, citing from Chok Yaakov (494:6).