Tosfos in Bava Kama 15, Gittin 88, Nidah 50a, Shavuos 29 ask: women can't be judges, so how did Devorah judge bnei Yisrael (Shoftim 4)? There are three answers:
1) it was kiblu alaihu - the people accepted her as judge. (This continues a point I made in Part 3, that you can accept someone as a judge who couldn't be appointed).
2) she only taught them the halacha, but did not actually judge
3) it was horaas shaah, at that time it was necessary to have Devorah judge, but otherwise a woman cannot judge. Maybe some people will understand this answer as an appendage to answer #1.
Woman as Mashgiach (in Kashrus)
Igros Moshe YD II 44-45 permits it because she doesn't have authority by herself, the authority stems from the rav hamachshir. Also, because of the aforementioned Tosfos, since Devorah was accepted she may judge, similarly, this mashgicha was accepted. My rebbi zt"l felt it was mutar without resorting to the reasons of R' Moshe, because many Rishonim say women (and geirim) cannot be appointed to Srarah positions that involve coersion (kefiya), but in the USA, a mashgiach has much less authority than that.
Women on Synagogue Boards
See Rabbi J. David Bleich in Contemporary Halakhic Problems II, Chapter 12.
Firing a Rabbi
Rabbi Gedalia Dov Schwartz, Shaarei Gedula, pp.118-120, and 370 n11.
See Rambam, Hil. Sanhedrin Ch. 25 for some laws relating to the people in srarah positions, and how they should be treated, and how they should act.
Some important responsa regarding these issues may be found in: Chasam Sofer O"C 12, Marashdam 85, Mabit ibid, Avnei Nezer Y"D 312. (These sources are quoted by R. Yitzchak Ralbag in his notes to Ritva, Makos 13a, published by Mossad Harav Kook. These are not easily accessible to me at this time.)
Let me end my discussion of Srarah by quoting the end of the last Ritva in the second perek of Makos:
Tangentially, we learn from our sugya (that someone who killed beshogeg and went to ir miklat and returned home after the death of the kohen gadol returns to the srarah position he previously held) that anyone whose forefathers held any appointment or srarah and he is fitting for that position, for the only argument here was when he because tainted due to his actions. It is not necessary to say that anyone who had a plain appointment without a term, that he may not be removed without a valid claim against him. I similarly saw in a responsa from Rabbeinu Meir regarding a Shliach Tzibur (aka Chazzan). And my teacher R. Meir would say in the name of his teacher, our master the Ramban that we learn this from the verse (Devarim 17:20) "so that the days of he and his sons will be lengthened among Yisrael," all appointments shall be for he and his sons.