"The nights are short in the summer."
Translation:Yazın geceler kısadır.
yazın is an idiomatic expression which means in (the) summer. it has nothing to do with the genitive ending -(n)In. therefore geceler is the subject of this sentence and it doesn't take -i ending.
if the nights belonged to the summer as you suggested the translation would be the nights of the summer are short. (yazın geceleri kısadır)
I have trouble telling when I should use a plural and when a singular form will be understood as a plural. I'm starting to think that if a word is the subject of the sentence, it needs "-ler/-lar" if it is plural, but that if a word is the object of the sentence, it doesn't need the "-ler/-lar" necessarily. Am I on the right track at all? :)
You are. But regarding objects... There is a definite and indefinite distinciton. A definite object(A definite object takes accusative marker.) needs plural suffix(-ler/-lar) if it is plural but an indefinite one doesn't.
EX: Ev temizlerim. (I clean houses.)<pre>
Evleri temizlerim. (I clean the houses.) Kitap yazdınız. (You wrote books.) Kitapları yazdınız. (You wrote the books.)</pre>
But sometimes indefinite objects take plural suffix too, for different purposes. But this is mostly used in spoken language.