I had a horse that couldn't be registered, and i still gave her a show name. I just called her cracker for short, and named her K.W.S.'s Firecracker for shows. Since she was born on the fourth of july. But yes! Totally give her a show name. It makes you sound more professional i think thank just a barn name. Just when you're filling out the paperwork for a show right down the name you want, which would be the show name you come up with. The people at the shows i went to didn't give me any problem.
I had a 14.1 hh Barrel Racing pony for years that did awesome and he was not registered and the people that had him before me called him Buster, So I made up a show name of "The DustBuster", He was a great horse and we showed on the larger barrel circuit for all of Atlantic Canada for years, I never had a problem at all.... So the answer is Yes, you can make up a show name if your horse is not registered as long as your not going to show a specific breed circuit like the AQHA shows and what not, then you would need a registered QH and so on.
No. They name the horse and then register
Most places require the horse to be registered to show it as well as having papers and they go by the registered name....
If your horse is unregistered, there is no way you can register it. Both sire and dame have to be registered. And some people give their horses nicknames that contain some of the registered name. Like my horse's name is Sheza Dakota Rose, we named her Dakota