I dont think there is any rules about that so long as they meet the level requirement they should be fine. Although I do think they cant cheer as the school they have to have a different name, and a coach. They couldn't be like the JBHS Wildcats (John Bowne H.S my old one.) They would have to be like the Aces or something.
They may not be able to depending on your state's high school bi laws. If it's the entire team and same coach I would be confident in saying it's probably not allowed.
Yes they did compete with their regular name there. The coach even had girls signing their own liability forms because she didnt hand them out earlier.
IDK if they'd be allowed to actually place, let alone use that win to qualify for the state championships.
But I don't see why a high school team couldn't enter an all-star comp toward the beginning of the season just to get the practice.