Is a gabai allowed to speak during chazaras hashatz, kaddish and/or kriyas hatorah for the sake of the tzibbur (e.g. to ask someone if they are a Kohen or Levi)?
The Mechaber says that only Sichas Chulin is not allowed. It follows that if the Gabbai has to ask them at that time then it is permitted. Unlike during Krias HaTorah where speaking is forbidden even when not for Chulin.
(See the Mishnah Brurah OC 125 sk 9 that says the Maharil wouldn't speak from the beginning of Kedusha until the end of HaKeil HaKadosh. It seems then that he did speak afterwards, so it must have been something important which would seem to be Muttar!)
In Aruch Hashulchan 124:12 he writes that a Rov who is asked a Halachik question during Chazoras Hashatz is permitted and obligated to answer and not wait until after Davening, because maybe the question is time sensitive.
In Hilchos Krias HaTorah 146:2 he brings a disagreement of opinions and concludes that the Rov should evaluate the situation and if it is time sensitive he should answer, otherwise he should hint to the questioner to wait.
In between aliyos, the custom is to permit talking when important. During kaddish, chazaras hashatz, and during the actual kriyah, I don't think it's allowed.
The Shulchan Aruch in Orech Chaim (124:4) says that the kehillah should be quiet and have kavanah for each and every brochah of the shaliach tzibbur. Commenting on that si'if, the Mishnah Brurah notes (si'if katan 17) that one should not learn during the chazaras hashatz, even if one has the kavanah to say amein to each brochah. Why? Because amei haaretz will learn from that behavior, thinking that they don't need to have kavanah during the chazaras hashatz, and will end up speaking "sichah bateilah". The Mishneh Brurah calls these people who cause other people to err "michati'in es harabim".
The same logic can be applied to a gabbai. If amei haaretz see a leader of the kehilah talking when people otherwise shouldn't be talking, even if that person is a gabbai and doing gabbai duties, kal v'chomer that that gabbai could risk falling into "michati'in es harabim". Amei haaretz will see him talking and assume they can too. A possible exemption could be kavod habrios, for instance if a non-religious family member is being given an aliyah and needs some coaching.
There's yet another instance where a Gabbai might speak during those forbidden times "for the sake of the tzibbur", and also FOR THE SAKE OF AN INDIVIDUAL! If the Gabbai finds someone speaking when forbidden, he is allowed to break that prohibition himself if he does so to stop that person from continuing his transgression. To stop an issur, one may trespass it for the sake of preventing the other from doing so. If a simile could help, it's like the guy in car 2 who follows the 1st guy in car 1 through a red light - to tell him he cannot do so. Well, not quite but almost like that.
I recall this issue from a gemora (Taanit 5b) where Rav Yitzchak speaks up during a meal to chastise Rav Nachman from doing so, since talking while eating is forbidden.