Sodom, Dhalsim, and Gouki by Syun Nishigaki, Adon by Setsuo Yamamoto, and Sakura and Gen by Tatsuro Suzuki are all listed in the liner notes of Rockman 2: The Power Fighters OST.
Nash and Rose from Zero1 don't apply here because Yuko Takehara doesn't return in Zero2. The Zero2 versions of those themes reflect that in that the former was made longer than Zero1's version and the latter had one segment (0:30-0:36) almost entirely recomposed. See for yourself.
Nash
http://www.youtube.com/watch?v=gtRPfkq29U8 - Zero2 Version
http://www.youtube.com/watch?v=npQSe4U17zI - Zero Version
Rose
http://www.youtube.com/watch?v=ikGVZSWO-D4 - Zero2 Version
http://www.youtube.com/watch?v=QPc0fsRsflc - Zero Version
EDIT (08/13): Ken's theme sounds like it was arranged by Tatsuro because it uses
that instrument. Anyone who listened to Sakura's theme will recognize it instantly.
EDIT (08/25):
-Opening is mentioned by Tatsuro in the liner notes on Booklet 8, Sagat, Rose, and Staff Roll from Booklet 9 to Booklet 10 [Last Tatsuro on 9 becomes first on 10], and Dan on Booklet 10 [the next time Tatsuro speaks].
-Zangief includes a sax, which is a distinguishing instrument for Setsuo during this time (i.e. Geminiman in PB/PF). Setsuo mentions these instruments on Booklet 8. [Said to have bass, guitar, piano, drum set, and brass/sax, and even a cowbell throughout the liner notes]. Drums are especially noticeable at the very end of Guy's theme. Vega generally sounds like Adon.
http://www.imdb.com/title/tt0417173/fullcredits#cast
EDIT (09/28):
-On Booklet 9, in the second Tatsuro section, he refers to Setsuo about Rolento and "Zangi;" this is merely Zangief shortened by two characters in order to save space, but it is definitely referring to Zangief. Chun-Li and Ryu both share a similar sound as Sodom.
-On Booklet 10, in the first Setsuo section, he mentions Nash.
EDIT (10/04):
-Booklet 7 uses similar words and sentence structure in the descriptions for Stage Gen and Stage Ryu VS Sagat. This tells me that Tatsuro handled the latter.
-On Booklet 9, Setsuo mentions "Sodom and Birdie," but since Sodom is definitely done by Syun, then so must Birdie. Use of Sitar (?) in Birdie's theme validates this.
-Since Tatsuro composed the Opening (i.e. main theme), it only makes since for him to handle all the other miscellaneous tracks that play off it [Player Select, Versus, Here Comes a New Challenger], just as Isao Abe must have done for Zero1.
-Game Over-Ranking Display, Stage End, Continue should still be Syun's.
Unofficial breakdown:
Syun Nishigaki:
Chun-Li,
Birdie,
Sodom,
Dhalsim,
Game Over-Ranking Display,
Stage End,
Continue,
Ken,
Gouki,
Cammy
Setsuo Yamamoto:
Zangief,
Nash,
Vega,
Rolento,
Guy,
Adon
Tatsuro Suzuki:
Opening,
Player Select,
Versus,
Sakura,
Ryu,
Gen,
Sagat,
Rose,
Dan,
Here Comes a New Challenger,
Sagat VS Ryu,
Staff Roll 1,
Staff Roll 2,
Staff Roll 3