What are the legalities concerning basing original work on the intellectual property of others?

It is possible that the comic-book characters are subject to trademarks which may be enforceable world-wide. This is frequently the case, and some characters (such as Batman) are the subject of numerous registrations for different classes of goods and services. A member of the consuming public could be confused into believing that your screenplay is endorsed or sponsored by the company that owns such trademarks, which would be a classic trademark infringement fact pattern. You should probably not proceed unless and until you have acquired the rights to use these names. You should not assume that the absence of a response constitutes consent.