According to the patent office, "laws of nature, physical phenomena, and abstract ideas are not patentable subject matter." If the theorem is embodied in a new useful process or device, that process or device could be patentable, provided the application meets all the other legal requirements.

Yes user interfaces are patentable.

Yes, some ideas can be patentable, but not all ideas. Some ideas might be better protected by trademark or copyright protection, while others might not be protectable at all. For example, this blog post explores the basics of what differentiates patentable and non-patentable ideas and gives suggestions on how to determine whether a given idea might be patentable:

patentsdemystified.com/are-ideas-patentable/

Shakespeare's works may have been protected by some primitive form of copyright back in the day, but it is all public domain now. None of Shakespeare's works are patentable subject matter and all of his works predate the creation of the USPTO. No, none of "shakespeare patent".

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