Does MarkStein Publisher run on the new macOS 10.15 Catalina?

Yes, MarkStein Publisher from version 6.5 is running on macOS 10.12 to 10.15. MarkStein Publisher from version 7 is running on macOS 10.13 to 10.15. Older versions up to 6.4 are running on macOS 10.11 to 10.14.