Mishna Berura 660:1:3 says that when the Sefer Torah is on the Bimah those on the Mizrach (eastern wall) turn around to face the Torah their right side is now facing Tzafon (north) therfore they start in that direction.
This is a minhag based on tashbisu (destruction).
The Mishna Berurah writes that it is a good practice to burn one’s
Chometz together with the Hoshanos from Sukkos--since one Mitzvah was
done with it, a second Mitzvah of Tashbisu should be performed with it
as well. The Rema brings a different Minhag--to bake Matzos with the
Hoshanos (Shulchan ...
Everything done in a Beit Kenesset is modeled on the procedures as they were done in the Temple, "HaMikdash", in Jerusalem. That is why a synagogue is known as "Mikdash Me'at", a small Temple (Yechezkel 11:16, Megillah 29a in the name of Rabbi Yitzchok).
In the Temple, when the first Kohen would enter in the morning to unlock the doors, they would turn to ...
The Nitei Gavriel perek 79:7 writes, that there are those that throw the aravot onto the aron kodesh. On the next line, he writes:
וטוב להצניעם עד עת אפיית המצות כדי להשליכם בתוך האש שאופין מצות, הואיל
ואיתעב בה, חדה מצוה ליתעביד בה מצוה אחרת.
"And it's good to hide them until matza backing time, in order to throw them into the fire, being that one ...
Yes, you should be holding them and even walk around a chumash placed on a chair or table(Ben Ish Chai parashat Ha'azinu 15; see footnote 2 in link 1 below).
It's not absolutely required, but an option available in order to maintain tradition and add merit, in my opinion; and is a worthy practice since it isn't too difficult(see link 2).
Links in hebrew:
In addition to sabbahillel's fine answer, I'd like to add something:
The Gemara (Shabbos 117b) brings a story of Rabbi Ammi and Rabbi Assi:
:רבי אמי ורבי אסי כי מיקלע להו ריפתא דעירובא שרו עליויה אמרי הואיל ואיתעביד בה חדא מצוה ליתעביד בה מצוה אחרינא
Rabbi Ammi and Rabbi Assi, when they came across the bread of an eiruv, they said a blessing over it....