ZERIZIM MAKDIMIM L'MITZVOS [Zerizus:Mitzvos]
(Beraisa): "Uva'Yom ha'Shmini Yimol Besar Orlaso" - one may circumcise any time during the day, but zealous people do Mitzvos right away - "Va'Yashkem Avraham ba'Boker"!
Rosh Hashanah 32b - Mishnah: On Rosh Hashanah, the second Shali'ach Tzibur (i.e. for Musaf) blows the Shofar. On days when we say Hallel, the first Shali'ach Tzibur says Hallel (after Shacharis).
Question: Presumably, the second blows the Shofar (to include latecomers in the Mitzvah) because "B'Rav Am Hadras Melech". For the same reason, the second should say Hallel!
We must say that the first says Hallel because Zerizim Makdimim l'Mitzvos - for the same reason, the first should blow the Shofar!
Answer (R. Yochanan): Indeed, the first should blow the Shofar. Our Mishnah discusses when there are royal decrees.
Tosfos (DH b'Sha'as, citing the Yerushalmi): Once, Nochrim thought that Yisraelim were blowing the Shofar (after Shacharis) to gather for war, and they killed the Yisraelim. It was enacted to blow after Musaf. Since this is after Shma, Shacharis, Kri'as ha'Torah and Musaf, it is clearly a religious rite and not for war.
Rosh (Rosh Hashanah 4:4): Even though the royal decree is Batel, the enactment remains until a Beis Din will annul it.
Question: If not for the decree, we would blow at Shacharis. But the Teki'os must be with the Berachos of Malchiyos, Zichronos and Shofaros, which are only in Musaf!
Answer #1 (Ba'al ha'Ma'or 12a): Letter of the law, every Tefilah of Rosh Hashanah has nine Berachos, including these three. Nowadays, the custom is to say them only in Musaf.
Answer #2 (Ran 9a DH Masnisin): If not for the decree, Chachamim would have enacted these Berachos in Shacharis to enable blowing the Shofar then, for Zerizim Makdimim l'Mitzvos.
Shulchan Aruch (OC 235:3): At night, l'Chatchilah one must say Shma immediately after Tzeis ha'Kochavim; the time lasts until midnight. If one transgressed and read after midnight he was Yotzei.
Beis Yosef (DH u'Mah she'Chosav Rabeinu v'Afilu): The Tur does not require one to read immediately, as long as he does not begin one of the activities forbidden before Shma. However, the Rif and other Gedolim require one to read once it is time.
Eshel Avraham (9): One who does not read immediately is not called a transgressor unless he begins a forbidden activity.
Kaf ha'Chayim (23): One who teaches others need not interrupt unless he started after the obligation to read began.
Shulchan Aruch (299:6): If one forgot to make Havdalah on Motza'ei Shabbos, he may make Havdalah until the end of Tuesday; some say that he must make Havdalah on Sunday.
Mishnah Berurah (16): All agree that l'Chatchilah he must make Havdalah on Sunday because Zerizim Makdimim l'Mitzvos.
Shulchan Aruch (433:1): If one checked (for Chametz) an Achsadra (a room without walls) by sunlight, this suffices.
Mishnah Berurah (6): This is if one did not check at night. Then, one must check in the morning, because Zerizim Makdimim l'Mitzvos. He may not eat before checking.