Is the villa considered a gift to the mother or part of the inheritance, especially since the father did not take possession of the gift and continued to reside in it until his death, despite the gift being documented by a notary public and witnessed by the children?

What the father did is not an immediate gift; rather, it is a bequest to an heir, and therefore it is invalid and does not take effect unless the rest of the heirs approve it. This is because ownership in a gift is not established by the contract before possession, just as a husband's gift of his residential house to his wife is not valid unless he empties it of his belongings and moves out of it. This is based on the hadith (definition): "There is no bequest to an heir." Therefore, the aforementioned property is to be included in the inheritance and divided among all heirs, unless they all agree to leave it to the wife, provided they are adult and of sound judgment.