Numerical aperture (NA [sorry, my mistake]) is the angular size of the entrance pupil from the object plane. It is an object space factor. f-number exists in the image space.
Originally Posted by BetterSense
f-number can be thought of two ways: the focal length divided by the entrance pupil or the angular size of the exit pupil from the image plane. An f/5.6 aperture presents the same angular light cone to the image plane regardless of the focal length. This is why depth of focus (the tolerance at the film plane) is only dependent on the f-number.
Likewise in the object space where the sample is, depth of field is dependent on the numerical aperture. Just like the f-number, the angular size of the entrance pupil is dependent on the dimension of the aperture and the distance to the object plane. You will find that each one of your scenarios having the same DoF will have the same numerical aperture.
Numerical aperture also is related to the resolving power at the object plane, just as the f-number is related to the resolving power at the image plane.