Learn Russian: Кот долго ловил эту мышь. - The cat chased that mouse for a long time.

The verb ловить means "to try to catch" or "to hunt." Because it is in the imperfective aspect here, it focuses on the ongoing process rather than a completed result. Since the cat spent a long time trying to catch it, "chased" is the most natural English translation. If the cat had definitively caught it, you would use the perfective verb поймал.

Why does мышь stay exactly the same in the accusative case, but эту changes?

The word мышь is a feminine noun that ends in a soft sign (a 3rd declension noun). These nouns do not change their endings in the accusative case. However, its modifier эта (this) must still agree with its feminine gender and accusative case, so it changes to эту following standard rules.

The English translation says "for a long time," but where are the Russian words for "for" and "time"?

Russian naturally captures that entire idea in a single adverb: долго. You do not need any prepositions or the word for "time" (время) when expressing how long a continuous action lasted in this way.

Shouldn't "that mouse" be ту мышь instead of эту мышь?

Strictly speaking, yes: эта means "this" and та means "that." However, Russian speakers use эта much more frequently and loosely than English speakers use "this." In many everyday contexts, it functions more like the word "the" or "that" to point to a specific noun the speaker and listener are already aware of.