I only know the answers to sum of your questions but Im sure others will fill in the gaps.
It generally isnt recomended take part in agility till the dogs are 1 year old and their joints can take it. However there are exercises you can teach a puppy that will come in useful for agility. For example, tunnels, running along side you (not under you), waits, sit stays, down stays, back end awareness exercises and directions. Just make sure you dont over do it and keep in to 10 mins sessions whilest they're young. You can also put in a bit of work socialising them to the compeition enviroment.
Workers or show cockers can take part. My workers love it. The guy I brought my second dog off said he's had several requests to breed dogs for agility and they are performing very well. There are a good number of show cockers at our agility class as well who are doing well.
I dont belive they have to be KC registered to compete but if its like flyball you do have to register as a competite but you do have to join the activities register
Good luck