"There are a lot of balls."
I read elsewhere that -들 was only really appended to people/animals/animate objects. Is the ball just counted as an animated object or is the rule incorrect or?
It's true this sentence sounds more natural without the suffix -들, but it is actually more complicated than that. Way more complicated. The suffix can sometimes be used even after uncountable nouns. It is totally optional in this exercise, but it's correct either way.
First of all, if plurality is implied within the sentence (like this exercise; there are many balls), you can omit the suffix. Second, as you said, the suffix is often omitted after inanimate nouns. Third, however, the suffix is necessary when the noun is preceded by 이, 그, or 저. Chances are the balls are spread all around, so the speaker wanted to emphasise their plurality.