Is a woman permitted to put up a mezuzah? If yes, does she make the bracha as well?
Women are permitted to put up a mezuzah, and would make a bracha on the mitzvah.
See Shevet Halevi 2:158, cf. Tshuvos Vhanhagos 4:238:6 who holds one should be stringent for the opinions that since a woman is unfit for writing a mezuza she should ideally not affix one as well.