Multilooking is not mandatory. It fixes the inconsistent pixel sizes caused by the angle of the incoming signal and produces an image with “squared” pixels. As a nice side effect, speckle is reduced to a certain degree.
When you cange the number of looks in the module you see the changes in your total resolution. If your data has a good resolution, TSX for example, you can apply multi-looking to reduce it to about 10 meters and still have sufficient resolution for most applications. However, if your data resolution is already coarse or if you need the spatial resolution for the search for certain objects I would personally advise to skip the multilooking.
As you said, the final geocoding is performed in the RangeDoppler Terrain Correction module at the end of your pre-processing. You can apply it on both multilooked or non multilooked data and (besides of some resampling) it won’t change your resolution any more.
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