Imagine the black lines should be cut along the edge which is now the red rectangle and the parts outside the edge should simply vanish without causing anything else (= no new lines, no hidden parts, nothing vanishes splits nor moves inside the edge).
Unfortunately original colors will be lost if there's more than one.
The first steps are the same as suggested in an earlier answer: Combine the black curves (Path > Combine), send them to back and duplicate the rectangle. If there's many overlapping curves you can select all and remove the rectangle from the selection by holding the shift. Do it before duplicating the rectangle.
Select the combined path and one of the rectangles. Apply Path > Cut Path. The selected rectangle vanishes:
Every black line below the red line is cut. You can select easily the unwanted parts and press DEL. Here's the result. The duplicate rectangle is moved aside and one of the closed curves is selected to show it's still one piece: