MITZVOS WE DO NOT DO BEFORE SUNRISE
(Mishnah): The following should not be done before sunrise, but if they were done after dawn, it is Kosher:
Kri'as ha'Megilah, circumcision, Tevilah, sprinkling (Mei Chatas), and Tevilah of a Shomeres Yom k'Neged Yom.
20b - Mishnah: The following may be done any time during the day:
Kri'as ha'Megilah; reciting Hallel; blowing the Shofar; taking the Lulav; Tefilas Musaf;
Korbanos Musaf; Viduy over the bull; Viduy Maaser; Viduy of Yom Kipur; Semichah; Shechitah; Tenufah; Hagashah; Kemitzah; Haktarah; Melikah; Kabalah; Zerikah; giving a Sotah to drink; Eglah Arufah; Taharas Metzora.
Yoma 28a (Beraisa): "Uva'Yom ha'Shmini Yimol Besar Arlaso" - one may circumcise any time during the day, but zealous people do Mitzvos right away - "Va'Yashkem Avraham ba'Boker".
Rambam (Hilchos Milah 1:8): We circumcise only during the day, after sunrise, whether on the eighth day or afterwards. If one circumcised after dawn he was Yotzei. Even though the entire day is Kosher, Zerizim Makdimim l'Mitzvos.
Shulchan Aruch (OC 58:2): If one did not read Kri'as Shma before sunrise, he should read it as soon as possible.
Mishnah Berurah (10): One should say Shma as soon as possible, but not if this will necessitate praying without a Minyan or saying Shma without Tefilin.
Shulchan Aruch (588:1): The Mitzvah of Shofar is during the day, after sunrise. If one blew Shofar after dawn, he was Yotzei.
Machatzis ha'Shekel (1): We wait until sunrise because not everyone knows exactly when dawn is (Rashi Megilah 20a DH v'Chulan).
Mishbetzos Zahav (652:1): The Levush says that mid'Oraisa, one should blow after sunrise. Day begins at dawn, but it is not fully day until sunrise. However, we may slaughter the Tamid before sunrise (Yoma 28a). This is like Rashi; Zerizim slaughter it, so we are not concerned lest they err (Rashi Pesachim 4a DH Nivdok).
Shulchan Aruch (652:1): One may take the Lulav any time during the day. Zerizim Makdimim l'Mitzvos and take it in the morning. The time is after sunrise.
Tur: One may take the Lulav any time during the day. Zerizim Makdimim l'Mitzvos and take it in the morning; the time is after dawn.
Question #1 (Beis Yosef): L'Chatchilah, the Mitzvah is after sunrise, like all Mitzvos that apply during the day (Megilah 20a). B'Di'eved, after dawn one was Yotzei!
Question #2: Regarding Shofar, the Tur writes that l'Chatchilah it is after sunrise!
Answer #1 (to both questions - Bach DH u'Mah she'Chosav she'Mitzvaso): Indeed, the Tur agrees that l'Chatchilah it is after sunrise. He merely teaches that the Mitzvah applies from dawn.
Answer #2 (Taz 1): The Tur teaches that mid'Oraisa, the time is from dawn. It is mid'Rabanan to wait until sunrise.
Answer #3 (Chachmas Shlomo 588:1): A Mishnah (Megilah 20b) lists Shofar and Lulav among Mitzvos that apply during the day, but the Mishnah on 20a did not list them among Mitzvos which l'Chatchilah are done after sunrise. Because Zerizim Makdimim l'Mitzvos, people are frenzied to do Mitzvos as soon as possible. We are concerned lest one do the Mitzvah before dawn. Regarding a Mitzvah mid'Oraisa (or from Nevi'im, which are like mid'Oraisa, e.g. Megilah), if we are concerned lest one do the Mitzvah too early and not fulfill the Mitzvah mid'Oraisa, we enact to wait until sunrise. But if he will probably repeat the Mitzvah later, we do not enact to delay the Mitzvah just to avoid a possible Berachah l'Vatalah (before dawn), which is forbidden only mid'Rabanan (according to Tosfos Rosh Hashanah 33a DH Ha).
There is no enactment to delay Lulav, for meticulous people hold the Lulav the entire day, and even common people take it several times. The Mishnah in Megilah does not decree to delay Shofar. It holds that Nashim Somchos Reshus, and likewise one may blow a Shofar needlessly on Yom Tov. People enjoy blowing the Shofar, so they are likely to do so again. The Tur forbids to blow a Shofar needlessly; hence he decrees to wait until sunrise.
Note: Alternatively, perhaps the Mishnah does not decree to delay Lulav and Shofar because it is expected that people will repeat them, during Hallel and Musaf, respectively. The Tur is stringent about Shofar because some people pray without a Tzibur and omit Musaf (they do not know it, and they do not have written Tefilos)