Do the children of a son who died before their paternal uncles and aunts inherit from these paternal uncles and aunts, knowing that some of them had no children and others bequeathed their entire inheritance to one daughter?

The nephew does not inherit if there are full siblings or paternal siblings, because siblings block the sons of siblings due to their closer relation to the deceased. The inheritance of the uncle goes only to his living siblings. He does not inherit from those who died before him, nor from the living sons of his siblings, unless the sisters are females only. In that case, the sons of the brothers inherit what remains by ta'sib (residuary heir) after the sisters' prescribed share.

As for the aunt writing her properties over to her daughter, if it is a complete gift and the daughter has taken possession of it, then it is valid as long as she is the only daughter. However, if it is merely a written arrangement for the properties to devolve after death, thereby disinheriting the remaining heirs, then it is not valid and is considered a bequest to an heir. It is not permissible unless the other heirs approve it.