Learn German: Die Katze schläft immer. - The cat is always sleeping.

The verb schlafen (to sleep) is a strong verb. This means it changes its main vowel for the he, she, and it forms. Here, the a gets an umlaut and becomes ä to match Die Katze (she/it).

The translation says "is sleeping", but doesn't schläft just mean "sleeps"?

German doesn't have a separate "-ing" form for actions happening right now. Die Katze schläft translates to both "The cat sleeps" and "The cat is sleeping" depending on the context.

How would I use nicht to say the cat is NOT always sleeping?

You would say Die Katze schläft nicht immer. The word nicht goes right before immer to specifically negate the "always" part of the sentence.

Could I say Die Katze immer schläft like in English?

No, the verb in a standard German statement must be in the second position. Die Katze is in position one, so the conjugated verb schläft must come immediately after it, pushing immer to the end.