Learn Polish: Dziecko ciągle pyta: dlaczego? - The child constantly asks: why?

Ciągle can mean both "still" and "constantly" or "continually." You can usually tell from the context. Because asking "why" is a repeated action that happens over and over, "constantly" is the most natural English translation here. If you wanted to unambiguously say "still," you could use wciąż or nadal.

Why is the verb pyta (imperfective) used here instead of the perfective form zapyta?

The word ciągle (constantly) shows that this is an ongoing, repeated, or habitual action. In Polish, repeated or continuous actions always require imperfective verbs. The perfective zapyta would mean the child will ask "why" just one time in the future.

If I want to add "me" to the sentence (The child constantly asks me), what case should I use?

You need to use the accusative case with the verb pytać (to ask). So, you would say Dziecko ciągle pyta mnie. This is a common trap for English speakers because with verbs like mówić (to say/tell), you use the dative case (mówi mi). But you always "ask" a direct object in Polish.

Could I use zawsze (always) instead of ciągle?