At our shows in this area, the judges don't do the "walking" part of the competition. I would actually like to see it done, though.
D'anvers would be a great breed. I have a pair. I get such a kick out of the little rooster strutting his stuff and my hen is as tame as a kitty.
We all have our preferences. I think my little Modern hens are the cutest things ever (except right now because the rooster has picked the feathers off their necks and they look like Naked Neck/ Modern crosses. Poor things!) I like d'Uccles, too, though. They're very cute.
Just one hen? Chickens do much better in groups. If you get bantams, you can keep about 3 in the same space you could keep 1 large fowl. They're easier to handle for kids, too.
Modern Games would be great. They do well in hot climates.