1. "Battle" is a pretty broad term for what we're seeing in this clip. This is a lot more playful, so the so-called battle is a lot more subtle. As some have said, yes, Ginger is largely following Fred's lead. But there are little moments where she inserts her own steps into the routine, separate from what Fred is doing. The biggest, most obvious gender-bending twist is Ginger's manly garb, as well as a more manly swagger in some of her moves.
2. Only five years into movie musicals, there is a lot more polish in "Top Hat" than what we saw in "Broadway Melody" or "Hallelujah!" This scene,