Translation:What is inside the box?
I am not sure whether は is required, but it does sound better.
To my understanding, に alone would just be 'in'. So 箱の中に何がありますか really is a straightforward "what is in the box?". Adding は to something turns the preceding part into the topic of the sentence, with が denoting the actual subject. That is more like "about the inside of the box (箱の中には), what is it?" Either translates to "What's inside the box?" though.
From what I learned in my class, に and で can replace は in Japanese's grammar (where it doesn't with も). Basically, the topic becomes implied by the use of either of these particles. It's not farfetched because Japanese uses context for the topic all the time. Using ある can mean "there is" or "I have" depending on the context or if 私は is present.
I'm assuming that it's in the same idea that adding it anyway adds emphasis to the topic of the sentence, since that is what は is for. What I'm confused about is why が wouldn't be used for this purpose since that's what が exists for when not paired with です/だ.
Yes, "Is there anything inside the box?" requires a different translation (as LeakyJam said). And はこの中には何ですか is grammatically incorrect; "to be" in the sense of "to be [somewhere] / to exist" is either いる（います）or ある（あります）, depending on whether it's alive or not. です is "to be [something]".
You would usually get that from context. Or if someone asked "すべての箱（はこ）の中（なか）には何がありますか？” that would mean what is in all of the boxes. Although the word "はこ" cannot be pluralised, because the word "すべて” meaning "all" is mentioned, you can gather that there are multiply boxes.