1. Levi wouldn't testify because he didn't hear it directly from Beis Din. Sounds like therefore what he'd say would be eid mipi eid, and that to inform others about BD's decision follows regular dinei Edus. See later on 25b mishpat. And see later 22a Ula said "kidshu leyarcha bemaarava. Did he hear it from BD?
21b. Bikesh Koheles. Perhaps we can explain this agaddata as follows:
a) Koheles wanted to reach 49th shaar bina like Moshe - perhaps Koheles wanted all of his works to be part of Tanach, just as everything Moshe said was automatically Torah (Rav Chayim, based on Bava Basra 14 Moshe kasav sifro v'sefer Bilam). Lo kam navi od kiMoshe - Shlomo, your works are not automatically part of Tanacha. (Only 3 of the 1,000 are in Tanach).
b) Koheles wanted to judge based on his gut feeling. See Makos 23b - Ruach hakodesh when Shlomo judged the 2 women and threatened to cut the baby in half. I think that Ruach Hakodesh meant Shlomo judged based on what he felt the facts were, based on psychology or human nature or call it what you want, even though there was no eidus. (See Rambam Sanhedrin 24:1 and Tur CM 15 and Beis Yosef, from Teshuvas HaRosh for similar dinim.) And Raavad in Hilchos Lulav about hadas sheniktam rosho means this - kfar hofia ruach hakodesh biveis midrasheinu - in his hasaga there. Shlomo wanted to even judge nefashos like this, and was denied that.
Moshe had din melech Zevachim 102 and Rambam Hil. Beis Habechira 6:1, I'm looking at Griz on Vayeilech; Rishonim at beginning of VeZos Habracha - vayehi bishurun melech (see this post http://shasdaf.blogspot.com/2008/12/yanai-hamelech-daf-66.html . In this week's parsha, Behaaloscha, perhaps the naar wanted Eldad and Meidad jailed and punished because they were being moreid bimalchus with their eidus against Moshe, the king, dying.
See also Chasam Sofer from Nefesh Harav __)
22a. Rashi bikuvya. The predecessor of dice were thigh bones. See Gods, Games and Gambling by F.N. David.
Ribbis. Kiddushin shiurim.
22b. Code to inform the chachamim of the Baytusim's plot. See Pesachim first perek about the goy who was eating from Korban Pesach and R' Yehuda ben Bava caught it "and his net was spread far away."