Using 4 (all driven or 2WD) traction wheels in a non-square configuration will cause the robot to skip around while turning, due to the fact that either the front or the back wheels can't slide on the carpet. True enough dependent on robot weight, drive train torque, wheel base, and wheel choice.
[*]Using 2 traction wheels and 2 omniwheels (all driven or 2WD) in a non-square configuration will turn freely, as the omnis can swing. True, the omnis will allow decent turning without side friction so the hop in the above example will be eliminated.
[*]Using 4 driven traction wheels in a square configuration will allow free turning?
??Not sure what you mean as "traction wheels".
[*]Using 6 wheel drive somehow magically works? False. Those that find this drive satisfies their design criteria will have the center wheel slightly below the others. All things being equal this will artificially shorten the wheel base or when the robot is tipped so that the robot rests on just the center wheels the wheel base is reduced to zero. Again trying to eliminate side friction in the drive system.
[*]Using non-square traction tank treads somehow works
False. Tank treads do a great (perhaps superior) job of driving in two directions (FWD/REV). However, the side friction in turns is the greatest of the drives listed. Teams have lowered the center boggies (as described above on six wheels) in an attempt to reduce the wheel base, which helps. Wide arcing turns are possible, turning on a dime (zero turning radius) produces incredible currents and attempts to force the treads off the sides of the mechanism. Some teams have enclosed the tread which helps but I have seen these drives eat the treads or drive them against the sides of the mechanism in such a way that it jams the tread.
I suggest that teams prototype the drive they are considering while playing the game.