I'd like to have about an efficient way of cropping the selected curve at every point where it intersects with the unselected curve (see picture).
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You can achieve this result by using the Boolean modifier.
Select your large curved mesh, and add a Boolean modifier set to Difference.
This will make the curved wall follow the contours of the arch in the direction of the Normals. You may need to re-calculate them (Ctrl+N) to get the desired result.
Now if you apply the modifier, you can remove the arch mesh and you should end up with a completed single mesh with the desired arch shape in it.