Hello there and thanks for your questions.

1. It would seem from the Gemora Yoma (second perek) and the Rambam (also see Minchas Chinuch 389) that all this was decided by Payis only. In the case of Musfai Shabbos, the Ramabam (Temidin u'Musafin 4:9) rules that those Kohanim who got to do the Tamid (based on the Payis) did the identical Avodah with the Musaf also. This shows us that the Avodah can be distributed based on a Payis only. It is not clear why the Rambam does not suggest having a special Payis for the Mussfin. In the case of Succos however this was not an option, since all Mishmaros participated in the Mussafin. How did the other Mishmaros who did not participate in the morning Payis, decide who should get to do the work? The Rambam rules that on shabbos when a new Mishmar arrives, they have their own Payis for the afternoon Tamid. So we can conclude that a Payis is always necessary to determine which Kohen performs any Avodas Tzibur. If we can decide the issues based on an earlier Payis, we do so. If that is impossible, as in the case of a new Mishmar, they must do their own Payis. The only situation not covered is Musfei Rosh Hashonah, which also had a ram and an ox (done by the same Mishmar) requiring more Kohanim than those needed for the sheep of the Tamid (see answer 2). How were the extra Kohaim chosen? It would seem from the Mishneh la'Melech that the next Kohanim in the line at the Payis were chosen for the extra work.

2. To calculate how many Kohanim were needed for the olos, see both Mishnayos Yoma 26b. A sheep required nine, a ram eleven, and an ox twenty four. Add on to each, one Kohen for Shechitah and one for Zerikah. Regarding the Sa'ir, since it was a Chatas not an Olah, there are no Nesachim, and only the fats were offered on the Mizbeiach. In my opinion that would leave us with three Kohanim, one for Shechitah, one for Zerikah, and one for offering the fats on the Mizbeiach.

All the best.

Y. Landy