Quests can be done on any difficulty and on any stage that has the criteria. If you repeat the quest criteria in some stages, it will not satisfy the quest (you must repeat the feat in another stage, but you can still do it on any difficulty).

First Row
Use Y + B attack many times on enemies in a stage
Clear out all rooms of enemies
Attain berserk mode many times in a stage
End a boss stage in berserk mode
Second Row
End a boss stage with 1 life ticker
Kill a boss with 100 percent bloody sword
Do 20 manual combo hits (same stage OK, does not need to hit enemies)
Kill 100 enemies in a row (counter tallies on lower right)
Third Row
Finish different stages without taking a hit (not even from berserk mode)
End all boss stages very quickly
Do a long COOL (frame) combo (all hits have white flash and whoosh sound)
Punch hit 10 mud zombies without being hit
Fourth Row
Achieve VIOLENT combo rank many times in a stage
Use X + A attacks for Saki or Aya many times in a stage
Enter Berserk mode from COOL (frame) combo hits
Evade (without attacking) 10 to 30 enemy attacks without being hit
Fifth Row
Use evade counter-attack to kill all bosses
Kill 100 enemies in mission 5
Use X + A attack for Annna many times in a stage
Clear survival mode to stage 18, 36, and 56 (do it three separate times)

*(translated from Famitsu)

Completing the last right two quests on row 5 will unlock the corresponding six new wallpapers (switch on the title screen using LB and RB). The other quests will unlock hair, arm, top, bottom, and leg parts for Dresses Up mode.