Question about the article for "Frau"
"Ich esse mit die Frau", I was told this is wrong. Does the feminine article become "der" when Frau is the object?
Also, to help you understand why "mit/with" is always dative: if the woman were the direct object, it would mean you are eating the woman. Think of the sentence as: "I eat [something not stated but understood] with the woman." Then you will understand why "the woman" is dative.
The common prepositions governing the dative are aus, ausser, bei, mit, nach, seit, von, and zu. So use the dative articles after them. dem (m), der (f), dem(n), and den (pl)
With the preposition "mit/with" the case is always dative. In the dative der->dem die->der das->dem and plural-die->den. When something is the direct object, as you mentioned, the case is accusative. In the accusative only der changes (to den).