Would I be correct in thinking that this is only logical because the ultimate system, the universe, is closed by definition (as far as the input of energy is concerned, anyway), and thus any part of it would have to have its entropy maximized in the long run - the long run being the time left until the heat death of the universe?
It's not sure. We don't know whether the Universe is finite or infinite, and whether or not it makes sense to consider it as closed. We're not even
postiive that it isn't open, but it sure seems likely that it is.
Even in closed systems entropy can be rearranged to decrease in one part and increase (more) in the rest.
Could you give an example of that?
Sure. Consider a fairly large room, no windows, with 100% efficient thermal insulation surrounding it on all sides. The system is the room, the boundary is is its boundaries. In that room is an electric generator with a full tank of gas, a dewar of liquid oxygen connected through a vaporization heat exchanger to the intake air filter of the generator, and a refrigerator connected to the electrical output of the generator. A timer starts the generator. The overall entropy increases significatly, the entropy of the oxygen and the generator increases a lot, but the entropy of the contents of the refrigerator goes down.
Maybe you need a compressor and tank, too, to prevent the pressure in the room rising enough to kill the generator .. but I think you see the idea.