Does equos agricolarum translate to the farmers' horses? I'm confused how sometimes s' or 's words sometimes have are accompanied by the nominative, while other times they are followed or preceded by the accusative. Which should you use? Or are the rules fluid? Thank you!

The rules about the genitive don't change, but the other word(s) will be in whatever case is needed for its job in the sentence. Examples:

The farmer's horse will be sold.
In this case, "horse" is the subject of the sentence; therefore, it'll be in the nominative case. The word for "farmer's" will be in the genitive because it's possessive (s).
Equus agricolae... (Or pre-first-century: equos agricolae)

He sold the farmers' horses.
In this case, "horses" is the direct object in the sentence; therefore, that word needs to be in the accusative. The word for "farmers'" will be in genitive because it's possessive (pl).
...equus agricolarum. (Macron over the second "u" in "equus" for accusative plural, 4th declension.)