The Gemara on 76b discusses appointing a ger or son of a ger to a leadership position (srarah). (The word sar, officer is found in Breishis 32:29, sarisa, see Onkelos, and in parshios Vayeishev, Shemos, Korach (historer) and other places as well.) From our Gemara, if one's mother was a bas yisrael and father was a ger, he may be appointed to srarah, but in Eretz Yisrael they would not even let someone like this be in charge of weights and measures, and in Nehardaa he couldn't be in charge of the wells.
Some other positions of srarah include melech, kohein gadol (though this is not discussed that much, at least in what I've seen), dayan, nasi, the gabai tzedaka, possibly a rav or shliach tzibur and even less likely but possibly a member of a board of a shul.
A very famous Rambam (Melachim 1:4-5) states that a ger can't be melech or have any srarah including head of the army, head of water distribution, nasi, judge, etc. because of the gezeiras hakasuv "mikerev achicha tasim alecha melech" - any appointment may only be from your brothers (to exclude a ger). Note our Gemara quotes the beginning of the pasuk that the Rambam quotes - "som tasim alecha melech mikerev achicha," but the Haghos Hagra changes it to the same way the Rambam brings it. In the next halacha, the Rambam says a woman may not be queen (ie head ruler, but it's not a problem to be married to the king) because it says "melech" and not malkah. And other appointments in Yisrael we only appoint men. He does not say srarah here, he says masimos, which I translate as appointments. So we see a neither a ger nor a woman can have srarah positions.
A kohen gadol who became tamei and a replacement served in his absence, when the first kohen gadol can resume his duties, he does. Similarly, someone who killed beshogeg and had to go to an ir miklat, when he returns home, he resumes any srarah he had before he left. See Makos 13a and Ritva s.v. Mai. So it is not easy to remove someone from srarah.
The Gemara in Bava Basra 8b, when discussing the srarah of gabai tzedakah, says we don't appoint one person to srarah, it needs to be two people. (Side point - I think the word gabai by itself means nothing. It seems it is an Aramaic word "gava'ei," meaning "collectors of." So really gavaei tzedaka mean collectors of tzedaka. It therefore seems to me the word gabai was reinvented to mean someone in charge of giving things out - like aliyos, even though the gemara discusses gabai tzedaka as collecting money and food to be distributed to the poor, but not that they distributed it, in fact, Bava Basra 8b says the tzedaka was collected by 2 people and distributed by 3 people, so distribution and collection are two separate things, not two sides of the same coin.) How can that be - there is only one melech! If we see how the Rambam formulates it, though, it's easier to comprehend: "We do not make srarah over the tzibur in monetary matters, less than two people." He adds the word "bemamon," which is not in the Gemara Bava Basra, but it apparently is in Yerushalmi Shkalim 5:2, which is not in my mishna, but when the gemara quotes the mishna, it does include the word bimamon. (Also note the Yerushalmi learns the din from "v'heim yikchu es hazahav" by the mishkan, but the Bavli and Rambam don't quote any pasuk for it). So it seems there are two types of srarah - regular srarah and srarah in money matters. (Does this mean one dayan in dinei mamonos cannot be appointed, because that would be srarah bidinei mamonos, but if 2 parties accept one dayan he may judge, because it's not an appointment, as Sanhedrin 5 says? Maybe so.)
Here is what we'll be dealing with in future posts:
If the son of a ger & bas yisrael can be melech, why were they punished with the incident of Aggripas?
If a woman can't have srarah, how did Devorah judge bnei yisrael?