The first sentence needs AND. Whenever you write a list you need AND between the last two items. I have horses, cows, pigs, sheep, ducks, geese, and goats. The quesiton "How many of them" refers to all of them. There's nothing wrong with it, but it's not very specific as to how many of each animal there is.

It refers to all of them. Yes it's proper question. If you wanted to be more precise you could have said How many of each of them do you have?

The question is fine. Them refers to the list of animals you named

In this context, the word "them" refers to the horses, cows, and sheep mentioned in the previous sentence. It is a pronoun used to collectively represent the animals on the farm.