Is a paternal aunt's gift of part of her share in a villa to her sister's son valid, and must the sister's son execute the paternal aunt's will within the one-third limit after her death, or has this part become his property, and must he carry out what the paternal aunt requested concerning the allocation of a portion to his mother?

What your aunt did by gifting half of her share to your brother and the other half to you is a joint gift (hiba mushaa'), which is permissible according to the majority of scholars. A gift is not legally binding except upon possession. If the giver dies before the recipient takes possession, the gift becomes part of the estate. If you did not accept the gift or dispose of your share as owners would, then the gift is not complete and is considered part of the estate.

Your aunt's debts must be settled before her will is executed and her estate is distributed, including zakat and expiations, if any.

What your aunt mentioned about a charity and a gift to your mother is considered a bequest (wasiyya), subject to the following: 1. The bequest must be from one-third of the estate and should not exceed it unless approved by the heirs. 2. A bequest to an heir is not permissible unless the other heirs approve it. If your mother is an heir, then the bequest is not enforceable without the consent of the other heirs.