Rosh Hashana Daf 34-35

1.  See this shiur by my Rebbe Rav Ahron Soloveichik zt"l that discusses many topics from these gemaras, such as

• machlokes Rashi/Ramban about precedence to go to tefila or to hear shofar
• Rabi Abahu about unifying the minhagim
• Whether brachos of Musaf are asmachta or real drasha.

2. 35a - last line - Rabban Gamliel held shliach tzibur was motzi the am shebisados with tefila; and above that - R' Yehuda davened every 30 days. .  See Mishkenos Yaakov O"C 90 about the am shebisados.  See also Shabbos 11a, Rabi Akiva Eiger in the Gilyon Hashas asks on Tosfos (Shabbos 11a s.v. kigon RSBY) that those who Torah umnasan are patur from Tefila but must say Kriyas Shma; but Tosfos Moed Katan 9b kan says they wouldn't say Kriyas Shma either; and "Ranach discusses it is a Teshuva."  I think he means R' Eliyahu Mizrachi Siman 70, and it is not at all explicit.  I think he's saying Torah umnasan don't have to daven in shul or Tefila betzibbur, d it has to do with kavod to them, or to minimize their bitul Torah.  I think to take it further The Shliach tzibur can be motzi Torasan Umnasan also, because they are ones wit htheir learning, and the am shebisados are ones with their work in the fields.  So:
a) Kriyas Shma, a dioraissa, they have to say, but not Shmone esray, which is dirabanan, BUT they get kiyum from shliach tzibur.
b) I think we can say - Brachos __ R' Yehuda Hanasi only said first pasuk of SHma - Torasan Umnasan must say first pasuk of shma and nothing else.  My Rebbi explained on Brachos 2a (See Perach Mateh Ahron Tefila 3:6 (page 52) that the first pasuk of shma is a kiyum of ol malchis shamayim but all the parshios are a din of tefila.  So the Torasan Umnasan are fully exempt from the Tefila part of Kriyas Shma also, and only say the first paragraph.  This would also explain Tosfos - where he says they are required in Shma he means first pasuk, and when he says they are patur from Kriyas SHma and Tefila means patur from Tefila part of Sham but are still obligated in Kabalas Ol part of Shma.

I think R' Gamliel's din is based on Maamados (Taanis 26a-28+) but there are some things not clear - whether the shliach tzibur only would be motzi them for their korban din of tefila but not the rachamim part of tefila (which fits well with maamados) but then the question would be birchas Kohanim is part of Korban din, but Neilah has no korban yet there is birchas kohanim.