Does a neighbor sin by repeatedly asking for things by way of borrowing and not returning them? And is it obligatory to seek the husband's permission before giving anything from the house? And is it considered a sin to say, "I do not have [it]," in order to preserve the husband's money?

What your neighbor takes from you is of two types: either a borrowed item (whose essence remains), which must be returned, based on the words of Allah Almighty: "Indeed, Allah commands you to render trusts to whom they are due," and the words of the Prophet (PBUH): "Render the trust to the one who entrusted you." Or it is a loan (which is consumed by use), and its repayment must be expedited, based on the words of the Prophet (PBUH): "Whoever takes people's money intending to repay it, Allah will repay on his behalf, and whoever takes it intending to destroy it, Allah will destroy him."

As for you taking from your husband's money without his permission, it is not permissible, unless what he gives you is not sufficient for you and your child, in which case you may take what is customary, based on the words of the Prophet (PBUH) to Hind: "Take what is sufficient for you and your child, in a customary manner," or if they are minor items whose disposal without permission is customary.

Regarding your lying to her, you can use allusive speech if her requests cause you harm, for in allusive speech there is leeway from lying.