Grammatically correct, yes. Natural, no. I think you mean something like “why didn’t you take it with you in the first place?”

I would add quotation marks around your reply, to make it more grammatically sound. And I would heavily reword the whole thing to make it more natural. For example: My sister's about to go out, and she's standing by the door, asking ME to fetch her backpack. So I asked her "why didn't YOU bring it?" / "why didn't you bring it yourself?" There's a couple of variations on the ending. The first is slightly more basic language; the second is a slightly more mature & classy way of speaking. If I absolutely wanted to say something like "instantly", I would say "originally". Or, as some people have suggested "in the first place" is also fine, but that has a more juvenile & argumentative tone, so is not my preference. However, most people would not bother with this level of detail at all, because it's already implicit from the context. Hence my first suggestion "why didn't YOU bring it?" where it's already obviously that I'm referring to a time in the past when it was logical to have brought it. Tip: avoid saying "and" too many times in one sentence. Notice how I only said it once.

It is wrong and unclear. Better: My sister is standing near the door planning to go out. She asks me to bring her her backpack. I reply ‘Why didn’t you take it in the first place? Or just ‘Why didn’t you take it yourself?’ It would not be perceived as a real question requiring an answer. It’s a rhetorical question whose purpose is to complain about being asked to do something.

It's grammatically correct, but to sound more natural, you could say: "Why didn’t you take it right away?"

Your sentence is almost correct but could sound a bit more natural with a slight tweak. Here’s a smoother version: "Why didn’t you just take it with you?" The word "instantly" sounds a bit formal or out of place here in casual conversation. "Just" or "right away" fits better in this context. Another option: "Why didn’t you grab it yourself?" Both versions sound more conversational and natural.