I bet something like a Bronzong would have to levitate, otherwise it'd be all "CLONG CLONG CLONG" as it hopped along, haha. Although i doubt Game Freak would worry about such minor things.
Whoever is at the top of my party, so whoever I want to use in the next battle, basically. Postgame, probably Feraligatr or one of my favourite pokemon (Sandslash/Kingler/Espeon).
I hope you can choose which one of the 6 you want to follow you, regardless of which is first to battle. (And on another note, I do hope that you can turn that feature off altogether when you want, like others have already said)