I just got the idea for this type of topic. The rules are simple: Use only one formation throughout the level. You automatically switch to Flight on the bobsleds and trolleys, so they don't count as switching formations for these challenges. On levels where you can't switch out of Speed formation right away, switch to the necessary formation as soon as you gain control. The TBG is not allowed, since it requires switching formations. Death is not allowed, except for LJ-C, due to its complexity. In team battles, you must knock all three off the battlefield without having the foes fall through the platform at the start. When battling Metal Madness, you may use another formation to get all the possible power cores, but not to attack. When battling Metal Madness, if the teammate you're using becomes trapped, use the character it switches you to until you can free said teammate. You cannot, however, attack during this time period except at the Metal Overlord part. I'll be updating this post with times anyone gets. The top 2 times will be listed. If no name is next to a time, it's mine. The list below is complete to my knowledge. If anyone finds a way to complete other levels with the specific formation or finds a problem with one of the bosses being unable to be defeated using only the specific formation, let me know all the details.
Speed onlySeaside HillTeam Sonic: 3:27:79 ~ SJ (2nd, 3:27.83)
Team Dark: 3:33.63 (2nd, 4:07.49 ~ SJ)
Team Rose: 2:01.73
vidTeam Chaotix: 2:20.41
Grand MetropolisTeam Sonic: 3:22.68
Team Dark: 4:31.19
Team Rose: 3:29.18
Power PlantTeam Rose: 2:12.93
Casino ParkTeam Chaotix: 3:06.00
Rail CanyonTeam Rose: 2:11.66
Frog ForestTeam Sonic: 3:59.36
Team Dark: 5:03.33
Lost JungleTeam Sonic: 5:34.73
Flight onlySeaside HillTeam Sonic: 4:04.93 (lol not found)
Team Rose: 2:01.24
Team Chaotix: 1:38.79
Power PlantTeam Rose: 2:18.19
Team Chaotix: 6:05.64
Casino ParkTeam Chaotix: 1:56.56
Lost JungleTeam Rose: 2:02.16
Team Chaotix: 3:47.31
Mystic MansionTeam Rose: 2:25.98 (2nd, 2:47.39 ~ Super Azrael)
Power onlySeaside HillTeam Rose: 2:28.68
Team Chaotix: 2:56.46
Ocean PalaceTeam Rose: 3:08.19 (2nd, 3:40.53 ~ Super Azrael)
Casino ParkTeam Chaotix: n/a
Bullet StationTeam Rose: 3:02.96
Hang CastleTeam Rose: 3:20.76
Bosses:
Speed onlyEgg HawkTeam Sonic: 1:20.56
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Team OneTeam Sonic: 45.01
Team Dark: n/a
Team Rose: 9.28
Team Chaotix: n/a
Robot CarnivalTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Egg AlbatrossTeam Sonic: 1:41.83
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Team TwoTeam Sonic: 31.21
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Robot StormTeam Sonic: n/a
Team Dark: n/a
Team Rose: 2:45.28 ~ Sondow
Egg EmperorTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Flight onlyEgg HawkTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Team OneTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Robot CarnivalTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Egg AlbatrossTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Team TwoTeam Sonic: 7.26 ~ Pokemonmaster888
vidTeam Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Robot StormTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Egg EmperorTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Metal Madness10:48.69
Power onlyEgg HawkTeam Sonic: 38.36 ~ Super Azrael
Team Dark: 46.39 ~ Super Azrael
Team Rose: n/a
Team Chaotix: n/a
Team OneTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Robot CarnivalTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Egg AlbatrossTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Team TwoTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Robot StormTeam Rose: n/a
Egg EmperorTeam Sonic: n/a
Team Dark: n/a
Team Rose: n/a
Team Chaotix: n/a
Metal Madness10:53.86