Learn Polish: Liczę dwa konie i trzy krowy. - I am counting two horses and three cows.

When you use the numbers 2, 3, and 4 with nouns that are not male humans, their accusative plural form (used for direct objects) is exactly the same as their basic nominative plural form. So, even though they are the direct objects of the verb here, the endings don't change!

If I had two cows instead of three, would I say dwa krowy?

No, you would say dwie krowy. Polish has different forms for the number 2 depending on gender: dwa is for masculine and neuter nouns (like dwa konie), but dwie is strictly for feminine nouns (like dwie krowy). However, for the numbers 3 and 4, you use trzy and cztery for both of these groups.

Since horses are living animals, why don't they use special "animate" endings here?

In the singular, masculine animals do use a special accusative form (for example, widzę konia - I see a horse). But in the plural, Polish simplifies things! Masculine animals are treated the exact same way as inanimate objects. Only male humans (like boys or men) use the special plural number forms like dwóch.

How exactly does the singular word koń become konie in the plural?