Hey, /r/AskAnthropology would be a great place for questions like this! Anyways, animal sacrifice is an extremely common phenomenon worldwide, so there isn't one way to answer your questions. Rituals, location, and the people paying will all vary from culture to culture, and within cultures as well. Nevertheless, I'll try to answer this from a Korean perspective, particularly the dotje rituals in the village of Gwenaegi in Jeju Island, where the village patron Gwenaegitto is venerated.

Religion was a key component to village solidarity in premodern Korea, with each village having its own patron deity. For the stability and prosperity of the village, this god would have to be appeased through rituals and sacrifices - this is especially the case in Jeju, which is much more shamanistic than the rest of Korea, which has historically had stronger Buddhist influence). The dotje is one of these rituals.

The dotje is carried out by a family, sometimes for celebratory reasons (often a wedding) and sometimes as a request for supernatural aid when problems arise. Some families regularly hold dotje every three or five years. But a family cannot just decide to hold a dotje whenever they feel like it; they need to consult a shaman or a fortune-teller for the most auspicious date.

Three days before the scheduled date, specially knotted ropes are hung on the main gate to ward off evil spirits from entering and interfering with the ritual. A pig is killed before the ritual to prepare for the sacrifice. The shaman removes the ropes when the ritual finally begins, as to allow Gwenaegitto to enter the house.

Sacrificial victuals are set out on the floor; rice cakes, boiled eggs, pears, wine, oranges, apples, steamed rice, etc. There is a candle on each side. As the dotje begins, the candles are lit, and white paper is cut in the form of money. When the family brings the shaman some pieces of linen, the shaman writes the names, age, and date of birth of each family member.

The shaman first takes out his tools and invites the god into the house. Then the dead pig is brought out. The skull of the pig is first punctured, symbolizing the offering of the head to Gwenaegitto, and slabs of pork, each one from different parts of the pig, are placed on nine different trays (care is taken to avoid spilling fat on the ground). The pork is accompanied with a stew made by boiling the remaining pieces of the pig and adding gulfweed and rice. The reason for the stew is to offer literally every edible bit of the pig (there are stories of how the shaman suddenly called out that Gwenaegitto was angry; the family would find that they had hastily missed some minor parts of the pig).

After the offering of the pig, the shaman recites the origin myth of Gwenaegitto (among other things) and divines the future and fortunes of the family. After the end of the divination ceremony, the family burns the paper money and the linen to avoid misfortune. The shaman leaves the house, holds a shorter ceremony to appease the minor spirits and servants that are thought to have accompanied Gwenaegitto, and ends the dotje by reentering the house to hold a simple final ritual.

Forgive me for only answering part of your question, but I only have a minute.

The Greco-Roman and Jewish rituals are fairly different. You have to understand that while their temples have a passing similarity, as institutions they are very different: different understandings of priesthood, different rituals of sacrifice and different understandings of what sacrifice meant. On top of all of this, in both societies all of these elements changed over time.

In the Greco-Roman tradition, the priests who do the sacrifices are elites within the community. Priesthood is a title of prestige, which is why Augustus snagged the title of Pontifex Maximus. In Israel the priesthood was an entire class set apart to retain ritual purity.

In both cases, you have to separate community sacrifices from family/personal sacrifices. For families, the animal is either brought to the temple or purchased at the temple (hence the money-changers in the New Testament). For the Greeks and Romans, they seem to have understood sacrifice to be transactional: "I do this so that you will do your thing". For the Jews elements of covenantal theology and sin offerings dictated the need for sacrifice.