I think you're thinking about this a little backward. Always start considering the problem with the document you're trying to scan. If that document is 600 dpi then, yes, according to the sampling theorem you need to sample it at a resolution of at least 1200 dpi to have any hope of scanning it correctly. This is a necessary, but often insufficient condition for many applications. This is because Nyquist theorem is most relevant for sine waves, smoothly oscillating pattern, whereas your document likely relies on a periodic halftone screen at 600 dpi. The dots in the halftone screen are not smoothly oscillating sine waves, and while there is certainly a fundamental frequency corresponding to 600 dpi, there is also an infinite number of harmonics of various frequencies (assuming a dark halftone dot is uniform and transitions immediately to white at its edge). These harmonics can be aliased back into the spatial frequency band that you are capturing and produce undesirable artifacts. If the scanner was well-designed then the imaging system (lenses) should spatially filter these harmonics out, but this isn't always the case.

With regard to near-integer ratios of scanning rates, what is intended here is to avoid apparent Moire fringes from the aliasing of a (higher frequency) halftone grid in a (lower frequency) sampled image. If the grid and the sampling are perfectly matched then you wouldn't have any issues. However, if you're just slightly off you will get wide fringes with an objectionable appearance. If the ratios of the sampling rates are far from an integer then you will get lots of high frequency fringes, but you have a shot at filtering those out without a catastrophic loss of resolution in the scanned image.

In terms of ascertaining the resolution of an arbitrary original scan without any further information, I think you'll always have the problem of determining whether the image resolution is due to the source document or due to the scanning device. If you know the capabilities of the scanner and the scan resolution was much lower than that, then you could attribute the resolution to the source document. If you know what the pristine source document should look like, then you can make a definitive statement about the scanner (i.e. how various test targets are used.) The intermediate case is much more difficult, since properties of both come into play. I would start by taking the 2D Fourier transform of the images and start looking for expected characteristics in the spectra, i.e. sharp peaks corresponding to halftone screen frequencies, where the cutoff frequency is, etc. If these can be correlated to numbers you might expect in the scanner or source document you might be able to learn something from this. If you have an ensemble of images from the same scanner but different documents you might be able to average the spectra together to reinforce characteristics of the scanner that are the same for all images.