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What's the song in the intro? (also: "What's the song in the BRII Trailer?")
The piece of music is the "Dies Irae" section of Verdi's setting of the Latin Requiem Mass, and is the most memorable/familiar portion of Verdi's Requiem. A number of different composers created their own renditions of Requiem; Mozart's own "Dies Irae" was used in the trailer for the first BR movie.

What's the song playing when...?
Since answering this question would require spoilers, please follow this link to the Assorted Spoiler Questions page (and, as if you didn't know already, there are SPOILERS ahead).

BATTLE ROYALE II OFFICIAL SOUNDTRACK:
The soundtrack for BRII was released on July 16, 2003, and contains 21 tracks from the motion picture score as well as the theme song "Mayonaka Shounen Totsugeki Dan." Except for the familiar "Requiem" by Verdi and a Beethoven piano piece that figures in the plot, the BRII score doesn't have a selection of classical music classics like the original had, and relies primarily on a new score by Masamichi Amano. We do hear a couple of Amano's motifs carried over from the original, as well as slightly updated versions of some themes.

From Kiriyama in the battleroyalefilm forums, working from material from Daniel Wang, here's a translated tracklisting of the soundtrack:

1. The fight for tomorrow
2. The fight for tomorrow part 2
3. Requiem
4. Wild Seven Declaring War
5. The time to attack
6. Shikanotoride 3B
7. The countdown to terror
8. Injured teammates
9. Getting weapons, and the death of Shugo
10. Nanahara Shuya's fight
11. Message
12. Mines
13. The Justice of a Ruler
14. Encountering Tragedies
15. Friends Forever
16. Landing Plan
17. The song of the Warriors
18. Teacher and Student Part 2
19. The glory of the resistance
20. Farewell to the Piano
21. Epilogue
22. Mayonaka shounen totsugeki dan by Stance Punks

MAYONAKA SHOUNEN TOTSUGEKI DAN ("Midnight Boys Attack Team") (the BRII ending theme song):
A wistful, unpolished 7-and-a-half-minute punk ode that becomes an energetic rallying cry at the end, the song by the band Stance Punks seems to have as much to do with BRII as it does with "The Midnight Workers," the nickname for Kinji Fukasaku's loyal filmmaking crew. With earnest lines like "I charge at my dream; I charge at your heart," the song was said to be a favorite of Kinji Fukasaku himself.

The song is the tenth track on the eponymous first album by the Stance Punks , which is available at CDJapan and Amazon.co.jp.; it can also be found on the BRII soundtrack album.

A romanization and translation of the song are available on mad as toads, courtesy of the lovely and talented Mink.

The song can be downloaded from this little spot on angelfire: http://www.angelfire.com/film/mayonakashounen

What's the song that Shiori plays on the piano?
Mark at unheardbeethoven.org helped us out a great deal with the proper identification for the piece: It's popularly known as "Farewell to the Piano," and is attributed to Beethoven, but his actual authorship of the piece is doubtful. It's also known as "Glaube, Liebe und Hoffnung" ("Faith, Love, and Hope"), and is a waltz for piano in F major. Masamichi Amano made an arrangement that includes a string orchestra for the version heard in BRII. Since the work doesn't have an opus number and the authorship is disputed, the Kinsky-Halm catalog of Beethoven's works lists it as #15 in the Appendix.

Here's a link to a midi of the piece, courtesy of lvbeethoven.com.