A: Do you need butter? - when we omit the article we are referring to any butter/ or butter in general. B: Do you need the butter?- when you use the definite article we are referring something specific, in this case you would be referring to specific butter, perhaps the butter in the fridge or a specific brand of butter. I would probably ask her, "Do you need butter?"

Both are good, one is asking about the need for butter the other is asking about the need for the exact lump of butter you have, but to most people it means the same thing

Both are correct~! "Do you need butter" would be refering to butter in general, nothing too specific. You would most likey be asking this if you had no idea as to what the recipe consisted of and were wondering if it includes butter. "Do you need THE butter."- this implies your a referring specifically to the butter in the fridge

Without "the" is fine, because if she needs butter, it does not have to be the one in the fridge even though it will likely be used.

Both are correct. You can use "the" to refer to the specific block of butter in the fridge or just "butter" if you want to know if she needs butter in general (as an ingredient).