Relationship between sampling and number of pixels?
Yes, that's correct. It's a natural extension of one dimensional up and down Sampling. However, one can define lattices (arbitary shapes) for sampling images, so one could downsample more in a particular axis than the other. Essentially instead of a sampling period, we have a sampling grid. There are benefits of using different freqeuency grids. Of course all of this holds without aliasing only when the sampling theorem holds. Otherwise you get aliasing.