If I have a polygon and I select to edit the nodes and at one of the vertices select "break path at selected nodes", I get two nodes one on top of the other. However when I select one of the nodes by clicking on it, it seems fairly arbitrary which one gets selected. If the wrong one is selected I then can't move the other one out of the way. I can select both at once or the wrong one, but not the right one! Is there anyway to cycle through node selection when there are several directly on top of each other, or a way to move one out of the way, move or delete the other, then move the first one back to the exact same place?

I'd try tab to the next node, then shift-tab back to the one previous, or vice-versa.

You can use ctrl + arrow keys to move away and move back.