Image sensor resolution
Essentially yes, you do have a very broad understanding.  But a couple of notes: Nyquist needs a little more than 2X, so call it 2X + or perhaps 2.1X or 2.2X.  However, you still will see large moire fringes (aliased patterns between the pitch of the object to the pitch in the image plane).

How do you get the right lens? Well, magnification will be dependant upon Object distance, Image distance and the focal length of the lens.

You say that you "... have an image with 10 μm wide black and white stripes ..." which I assume means that you already have a lens in front of your imager, so what is needed in this case is that the magnification has to increase by 2X.  It is likely that you are dealing with a optical system that has de-magnification (objects are likely to be larger than 10 um - magnification is less than 1).  But the 2X (or 2.2X) can be applied.  In  general this means that you will change the focal length of your lense.