Can a public theatre refuse an idea for a sketch show based on the shows format?

Answer 1

yes- just because it's public doesn't mean there are no rules.

Answer 2

Yes. There can be many reasons why a theater, even a public one, would refuse a show. They have standards that they set, and have the right to choose the shows that they want to host. The fact that it is public, rather than private, usually means that they will actually have stricter standards, as they have to be responsive to public opinion!