Is there a specific term that is usually used to describe live poultry, or live chickens, or any type of live animal, which is for sale in a market and which the customer takes home alive to butcher later? I wondered if there is a term to distinguish between poultry which is taken home alive, versus poultry which is alive when the customer selects it, but which is prepared for eating at the market and taken home already plucked, gutted, etc.
活禽 or 活家禽 means live poultry, as @user3306356 answered. (Reference)
For living animal, it is the name of the animal with 「活」(alive) at the beginning. Such as 活雞living chicken、活鴨living docks、活豬living pigs、活牛living cows. Sometimes 「生」replacing 「活」 can also mean the same thing, e.g. 活豬 = 生豬, but 「生」have the meaning of "born", "give birth to". E.g. 母牛生小牛, 母雞生雞蛋, so depends on the context, be careful to use the right word.
For plucked, gutted animals, we would say its meat. If you want to say fresh meat, you can say 鮮肉, and specifically 鮮雞肉fresh chicken、鮮豬肉fresh pork、新鮮羊肉fresh lamb/mutton、新鮮牛肉fresh beef, etc.
If you want to say the dead animal, just add 「死」in front, like「活」、「生」. E.g. 死雞、死牛, etc. But we rarely refer their edible meat as dead animal.