"It is not hot at night."
Wondered the same but i think i got it: In "yoru ame ga furimasu" ,ame is the subject while in "yoru ha atsukunai desu" yoru is the subject. Not a pro at explaining but i think its safe to say when its a verb, no "ha" needed and when its an adjective "ha needed. Correct me if i am wrong.
It's not really the verbs/adjectives thing; but it is true: in よる雨がふります, よる determines the time but is not the subject of the sentence (there's 雨 for that as denoted by が). Whereas in this sentence, よる is the actual subject - for a more representative translation, think of it this way: "the night is not hot".
in Japanese there is no action taking place at night, you are just describing the night. I know it can be confusing because of the "at" preposition in the English sentence but で is not a translation of "at", で is used to mark a location in which the action happens or the means in which something happens.