Learn Russian: Кот поймал мышь.

It actually is in the accusative case! Мышь is a feminine noun ending in a soft sign (-ь), which makes it a 3rd declension noun. In the singular, 3rd declension feminine nouns look exactly the same in the accusative as they do in the nominative, even if they are alive (animate). If it were plural, you would see the animate rule apply: Кот поймал мышей (The cat caught mice).

Why use the perfective verb поймал instead of the imperfective ловил?

Поймал emphasizes the successful result of a single, completed action: the cat actually caught the mouse. If you used the imperfective ловил, it would mean the cat was trying to catch or chasing the mouse (focusing on the process), or that he caught mice regularly as a habit.

Why use кот here instead of кошка?

While кошка is the generic word for the feline species or a female cat, кот specifically means a male cat. However, in fairy tales, literature, and simple examples like this one, кот is very commonly used as the default subject.