Why did the anshei Yerushalayim hold the lulav all through the morning, but put it down/send it home when they went to the Bes Medrash?
Rebbi Elazar bar Tzadok said that the Anshei Yerushalayim would send their Lulavim home when they entered the Beis Midrash. I understand your question to be asking why they sent them home, when they could have simply put them down in the Beis Midrash. RASHI explains that they did not hold the Lulavim while they were in the Beis Midrash, because of the fear that they might get so involved in their learning that the Lulavim would fall from their hands. Why, though, did they send them home, and not merely place them done outside of the Beis Midrash in order to pick them up again and walk with them when they went home (which would have shown more love for the Mitzvah)?
In a simple sense, it could be that since they did not leave the Beis Midrash until after nightfall, the Mitzvah of Lulav was no longer relevant and there would be no display of love for the Mitzvah by carrying the Lulav at that time. Alternatively, they feared that by placing their Lulavim outside of the Beis Midrash, the Lulavim might become damaged from all of the people coming and going, or stolen by Nochrim, or the like.
RAV YECHEZKAL ABRAMSKY (in CHAZON YECHEZKAL, Sukah 2:12) writes that the reason they sent their Lulav home when they entered the Beis Midrash to learn is because leaving it on the floor for an extended period of time would be Bizuy Mitzvah, a disgrace to the Mitzvah. Leaving it on the floor for a short time, though (such as when they went to the synagogue and read the Torah, as the Gemara mentions), is not considered a disgrace to the Mitzvah.
The Acharonim have a different approach. They write that the reason the Anshei Yerushalayim held their Lulavim all day long was not merely to show their love for the Mitzvah, but because they maintained that as long as one is holding his Lulav, he is fulfilling the Mitzvah of Netilas Lulav. It is one long, continual performance of the Mitzvah, which does not end until the moment that one puts the Lulav down. Therefore, they were careful to hold the Lulav as much as they could, without setting it down, in order to make the Mitzvah last as long as possible. TOSFOS (39a, DH Over) writes that this practice of Anshei Yerushalayim was a "Mitzvah Min ha'Muvchar."
Now we can understand why they sent their Lulavim home when they entered the Beis Midrash. Once they were no longer holding the Lulav, the long act of the Mitzvah was over (and to pick it up again would be a second Mitzvah, and not a continuation of the first, and the Torah only requires a person to pick up the Lulav once each day). Therefore, they sent their Lulavim home. (RAV MOSHE STERNBUCH in Mo'adim u'Zmanim (2:116) brings proof to this from the Yerushalmi (end of third Perek), which says that while the Lulav is in one's hands from the first time that one picked it up during the day, one is considered "To'eh b'Davar Mitzvah" if he carries it out to Reshus ha'Rabim, since he is in the process of fulfilling the Mitzvah (this is in contrast to what our Gemara says on 42a). He similarly infers this from the words of the RAMBAM (Hilchos Berachos 11:5, 15).)
The ME'IRI, however, seems to contradict this approach, because he writes that when they would put down their Lulavim in order to read from the Torah or for Birkas Kohanim, they would then pick it up again afterwards.