Is it possible to say "La souris était mangée par le chat. -- The mouse was eaten by the cat."?

To your first question: yes it is possible. The difference is that 

... était mangée is imparfait and

... a été mangée is passé composé. 

Thank you. But when would you use the imparfait and when would passé composé? What's the difference in meaning here between the two tenses? (Assuming you were the cat.)

Sorry, I meant: "But when would you use the imparfait and when would you use the passé composé?"

Usage of imparfait vs. passé composé is a separate topic in itself which I can't answer exhaustively in a short post. There are several lessons on this topic on kwiziq, however. I suggest you either browse the Library or progress through the lessons until the start to appear naturally in your lesson plan.

Or, alternatively, if you're impatient, why not try Google?

Thank you, Chris.