Here's some notes on a few bands...
Vibra Prime (Team Baron - Cecil, Ceodore, Kain, Rosa, Cid) - there's no elemental aspect to it (despite "Holy" appearing)... overly physical based... hits all monsters equally (no solo monster bonus)
Final Fantasy - it's more affected by high magic defense (less on physical), non-elemental
Advent of Phoenix - Even if a target is resistant or can absorb the elemental (Fire), the monster still takes the damage based on HP... for some reason it deals an additional 10% more damage to the Blue Dragon... go figure.
Makeshift Cannon - loses 30% damage vs high magic defense targets, loses 50% vs high defense targets
Infinity - Holy elemental, loses 50% damage on high magic defense targets... only 5 character band to be absorbed by the Blue Dragon...
Frost Wave - Ice elemental (duh), 50% bonus damage to a solo target, Blizzaga level magic I think (no split damage loss)
Rapid Fire - affected by both Black Magic moon and Attack moon (Attack is more beneficial with two attackers)... it is affected more by physical resistance...
deals 25% less damage vs high magic defense, 60% less damage vs high defense... needs Limit Ring to expose damage potential
Constructing a team of Tsuiknowa, Izayoi, Ceodore, Palom, Leonora is alright, but are really limited to those two bands.
So, AFAIK, the only two Ninjutsu based bands to heal the Blue Dragon are Frost Wave and Flare Tornado.. but the "no physical blows" on the monster rule generally holds true. Frozen Moon Dance, Blazing Moon Ring, Saint Dive... and many others (including Rocket Launcher) are designed to by physical based (using the Attack command or other means). Although Rocket Launcher is not a physical blow, it actually combines the attacks of both Ceodore and Cid...

I believe that Throw and Shuriken are powered by STR (in addition to level). I noticed that Gekkou seemed to do more damage than Edge throwing the Fuma Shuriken... and the only real difference between Edge and Gekkou is that Gekkou has far more STR than Edge.