For some reason, on one side of the objects, nodes are being added correctly, in the middle, whereas on the other side the nodes are being added in odd positions. Am I doing something wrong? I've attached the file.

Interestingly enough, I've tried the same thing in Adobe Illustrator and got the same effect. I have no idea why this is happening.

Unfortunately I am not seeing that and I can't account for it! When I deleted the extra points and selected the two endpoints each subsequent addition added points in between each pair of points exactly as expected. I checked the Ins, Shift+I, and control bar and they all work.
If you are using the double-click method that would add a node at the point under your pointer

Very odd. I was using the plus button in the top left of the toolbar. I did select the path, not the two nodes individually.

Well, I worked around this by recreating the bars using a rectangle.

Thank you for your time.