What is the ruling on saying: "I ask You by so-and-so," and is this considered asking by other than Allah, and does its utterer apostatize and become obliged to perform ghusl and renew his Islam?

Supplicating to Allah through a specific person, such as a prophet or a saint, is not shirk (polytheism). Rather, it is an innovative form of intercession about which scholars have differed, with the preponderant view being that it is not permissible. Asking a creature through another creature like it, such as asking by the right of a father, is not shirk. Shaykh al-Islam clarified that if the statement of the Salaf, "I ask you by Allah and by the ties of kinship (Rahim)," is evidence for its permissibility, then its meaning is not an oath by kinship but rather on account of it, because kinship entails rights, similar to asking through righteous deeds, or through the supplication and intercession of the Prophet, peace and blessings be upon him. And as 'Ali ibn Abi Talib did with his nephew 'Abdullah ibn Ja'far, it is not an act of swearing but rather a right of kinship.