Translation:The publisher eats with the author.
It's a masculine noun. Das is used for neutral, and Der for masculine. However in this example, it is in the dative case, so both neuter and masculine articles change to dem.
It is masculine. Duolingo's dictionary is really poor. Don't rely on it.
If it is, it shouldn't. Although 'isst' and 'ist' are homophones, "Der Herausgeber ist mit dem Autor" is ungrammatical in German.