Ok, here's what I got from the 2nd vid (since my Japanese sucks and all =D), so you get the gist of it ^^
"Shinjitsu no Yukue" is a song from the first Bleach musical I think (who knows, I haven't seem em XD)
Gin randomly starts singing it after Hitsugaya gives Hinamori an order ("meirei", which is also in the song!
)
After a bit, Hitsugaya tells Gin to shut up XD (loved that part) but then someone says "meirei" again and Gin starts singing AGAIN XD
I didn't quite get the next part, but I think Hinamori went away 'cause she didn't understand...something? XD
Then Hitsugaya calls Kira and orders him to do whatever (after clearing up that Kira's a vice-captain and Hitsugaya's a captain, which is why he has to listen XD)
Yeah, repeat the "meirei" stuff then~ (at this point Hitsugaya is also pissed and leaves XD)
I don't get the end either, where Gin gets all close to Kira and stuff, but I know he tells him something and at the end he says it was a lie and leaves XD
Meh, this got long, lawl. And I just wrote that without watching the vid again, so don't wonder if the order's wrong somewhere D:
I hope it makes more sense now~