Does this have the same meaning as in English, in which the "they" are thinking about a decision overnight?
No, this only applies to sleeping on a bed or a couch. If you don't want to make a hasty decision and want to sleep on it, you'd say: "Ik wil er een nacht(je) over slapen." So we sleep "over" something, and don't forget "nacht" or "nachtje" in this phrase!
There is no real difference in this sentence, I suppose. It's a diminutive suffix so it's supposed to convey something smaller than the usual word. But we often use it interchangeably in Dutch. "Geef me een minuut" has the same meaning as "Geef me een minuutje" (Give me a minute).