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You aren't being paranoid; if those wrinkles affect the planarity of the corner points, then it may very well degrade the calibration result. But you'll never make the absolute perfect calibration target, so you really need to be thinking about how much it might be affecting the results. It's best if you start with your application requirements, and then work backwards to figure out how much accuracy you really need.

For example, the undistorted images you show look reasonable; clearly you're getting at least some decent distortion coefficients. If you only need roughly undistorted images, then you're already fine. If you need more accuracy than that, then decide how much, otherwise you may find you go down a rabbit hole chasing accuracy you don't need.

There's no reason to rule out paper a priori - you can certainly get good calibrations with paper targets. But you may want to take more care in mounting the paper. And in some environments the paper might, for example, absorb moisture and wrinkle. Again, depends on your application and usage environment.