No, it's always "die" in the accusative case. But native speakers would hardly ever missunderstand this sentence. The inverse word order (OVS) usually is only used in contrast situations (Was ist mit der Katze? Wer isst die Katze? - Die Katze? Oh! Die Katze! Ja, die Katze isst die Maus, natürlich.)... ;)
Der=The "Die" is by plural or singular. Plural= The girls/Die Mädchen Singular="Das" Mädchen
Singular="Die"Frau/The woman Plural="Die"Frauen/The women
Always use: "Der" by he "Die" by she "Das" by it
By plural you always use "Die" Die Mädchen/The girls Die Frauen/The women Die Männer/The men Die Buben/The boys Die Tiere/The animals,...
Yeah with the Die, Der, and Das things will eventually just start to work themselves out in your head. Just like you know "an couch" sounds wrong and "a couch" sounds right, you will slowly start hearing the same things in german. You will go to write "Das Mann" and be like "That totally sounds wrong" then go to write "Der Mann" and be like yes that sounds right!
Sometimes yes, but it depends. "Der Hund" can be both, "der Rüde" is a strictly male dog, and for a female dog I can only think of "die Hündin". Often, but not always, you can add -in to derive the female form of an animal (though sometimes you have to make the main vowel an umlaut).
That's a bit more difficult in German. There are 2 main past tenses, the "Präteritum" (preterite/imperfect) and the "Perfekt" (past perfect), which are "Die Katze aß die Maus." and "Die Katze hat die Maus gegessen.", respectively. Note that "essen" is an irregular (strong) verb.
Not really. In German we use the same verb form for "eats" and "is eating" (isst). So this German sentence can have both translations. Only colloquially you can say "The Katze ist die Maus am Essen." with "ist am Essen" meaning 'is eating'. But I don't think that the course will teach this form, as it's not (yet) standard.