3
You can use Live Paint groups to add the wanted anchor points.
You need the wanted shape and the offset version as separate paths with a stroke, but no fill. The lines should just be plain lines with a stroke.
Select the lines and make them a Compound Path. (Ctrl / Cmd + 8)
Position the shapes as wanted on top of the lines.
Select the shapes and the lines ...
3
You could use the Edit Paths by Nodes tool N to grab a path and bend it. There's no need to add any extra nodes to the rectangle for this to work.
Example
What this really does is pulls out Bézier curves from the nodes.
When you have the rectangle selected using the Nodes tool N, click and drag a rectangle selection around all four nodes, this will select ...
2
You can use "Bend" path effect as follows:
Select your path.
Path > Path Effects... and add Bend effect.
Select Edit on-canvas and use the mouse to bend the path.
1
I certainly understand a desire to have things like this more automated.. but then again, if everything can be done with a few menu choices and button clicks.. where's the craftsmanship? Some procedures merely require dedicated focus, that's what makes artists as opposed to merely mouse operators.
I suppose one could use Object > Path > Add Anchor ...
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