Who are the descendants of Rav Pappah mentioned in the text of the Siyum for a tractate of Talmud, and why are they so important that we mention them all by name? Why don't we mention any other great rabbis from that era?
Not all the "Bar Papa"s mentioned in the list are the sons of the Rav Papa from the Bavli.
The Sefer HaEshkol, written by Rav Avraham Av Beit Din (father-in-law of the Raavad) brings a statement of Rav Hai Gaon (and a proof) that the ten Bar Papas are not all sons of the Rav Papa the student of Rava: HaEshkol Hilchot Sefer Torah 14.
They did not all live at the same time and place, Rav Hai mentions.
He further says that the reason for listing the ten of them at the siyyum is that there is a tradition that saying them prevents forgetfulness. Therefore, when one reads a chapter and reviews it, he says that he will review it, and also says the mnemonic for, and the names of, the ten Bar Papas.
Nowadays, I've never heard of anyone saying the mnemonic.
Rav Tzvi Binyamin Auerbach, who wrote the "Nachal Eshkol" commentary on the Eshkol, comments:
ועתשו' רמ"א בסופה ויש"ש סוף ב"ק
for many further reasons for listing the ten sons of Rav Papa(s).
I'm not certain what the "teshuvat Rama" is here, but second reference is to the Yam Shel Shlomo, but it delves into esoteric concepts, and I am out of my depth.
One explanation (from Artscroll Hadran)
Rav Pappa was a very wealthy man who would make great celebrations each time he finished a Mesechta, and invited his 10 sons as well as many others. He therefore brought glory to Torah, which was "reflected in the scholarly attainments of his sons." So we honor Rav Pappa and his family by mentioning them at every siyum.
Another explanation (from here)
There are 10 places in shas where there is a machlokes, and Rav Pappa made peace by accepting both of the opinions. Since Rav Papa did this, he was a vehicle in bringing shelaymos (completion) of the Torah into the world. He therefore merited 10 children, and we mention them at the end of a mesechta because a siyum is a time of completion.
A third explanation (also Artscroll Hadran):
Rav Pappa symbolizes Moshe, and the names of his 10 sons symbolize the Ten Commandments.