How is the sacrificial animal divided? Is [every part] divided into thirds i.e. the liver into thirds, the head into thirds, etc…? How is it divided into thirds?

Sacrifice is a confirmed practice of the Prophet because he offered sacrifice and so did the successors, and because He said:
"Three are obligatory upon me and voluntary for you; sacrifice, and the two rak'ahs of Morning Prayer [En: Daybreak Prayers]".

Islamic law clarified how to dispose and divide the sacrificial animal. It is
recommended to divide the sacrificed animal into three thirds; one third is eaten, another third is given away as gifts, and the last third is given as charity. There is no harm if more than one third is eaten or given as a charity. This is because distributing the animal is recommended and not an obligation.

Ibn 'Umar [May Allah be pleased with them] said:
"Sacrifice animals are divided [into three thirds]; one third for you, a third for your family, and a third for the poor ".
The meat is [the part of the animal which is] distributed because it is the main objective, and is the part which benefits the poor and the needy. As for the innards such as the liver and other [parts], it is recommended to divide them, though there is no harm if they are not. The head is not distributed but belongs to the person offering the sacrifice— it must not be sold or given to the butcher as payment.

It is obligatory to sacrifice an animal if one has vowed to do so and the person offering the sacrifice may eat from it and give it away as gifts and charity to the poor and the needy.

Al-Qadi abu al-Tayib states that the Hanabli school maintain that it is impermissible for the owner of the vowed sacrifice animal to eat from it, which is the apparent meaning of the words of imam Ahmed.
The reliable opinion for this fatwa is that it is permissible to eat from the sacrificed animal. This is because vows follow prescribed rulings and the prescribed rulings concerning the legal conditions of the sacrificial animal is that it is slaughtered to be eaten. The fact that the animal is slaughtered in fulfillment of a vow does not change anything except being an obligation.
Allah the Majestic knows best.