I would add that in general people have been very kind about answering questions, even fairly stupid questions
and that each of the shows my daughter has attended have had different procedures. At these shows, no health paperwork was required, but there is a poultry inspection before any bird is allowed inside the show grounds.
With your runner duck, you may face a slightly different show procedure, because if I understand correctly, some - but not all - shows will set up a pen so that the judges can see the ducks run.
Most shows provide initial bedding. For some shows, especially a county fair, your birds may have to stay several days. Just like at a horse show, I'd encourage you not to go too far, because your birds can run out of water or otherwise need attention. You'll want a food and watering container that they can't knock over, and I suggest bringing tape and twist ties and other extra fastening doodads in case you spend the day with your birds discovering new and exciting ways to tip them. You'll see many solutions for that at the shows.
It's typical for cages to be assigned, and to be assigned in such a way that the birds in the same classes are adjacent to each other. Thus, if you brought several breeds, they could be in different areas. The judge handles each bird from the cage. At this time, you should stay away from the judge and his scribe to give them freedom to work. Judges sometimes write comments on the cage tag, especially for severe faults.