Look at the premium list and it will tell you. At a minimum, fully feathered. The judge has to be able to see the markings and color.
My local fair says no birds under 6 months. Each fair will have it's own rules. The APA might have an age rule that applies across the board, or the individual shows might be able to set a limit. I don't know the answer to that one.
I'm not aware of an APA rule of showing age however most will not show any stock until the are between 5-8months of age. I've shown a pullet at 5 months and she won Ch. AOCCL
The timeing for your breeding for Pullets and cockrells will be based on breed, bantams often do fine as pullets around teh 6 month mark, you want them right before they brgin to lay. bantam Cockrells around 10-11 months. Any bird 12 months and older is a hen / cock. cockrells and pullets are less than 12 months of age.