What is the best file format to store gel images?
Always keep the raw data files! This is always a good idea for scientific data, and the only exception should be if the raw data is prohibitively large and it is not feasible to store it completely. This is not the case for gel images, so I would always keep the originals, and then use a cropped and edited copy in a suitable format for documents.

The original image has the full amount of data, any conversion can potentially destroy data. For example, some imaging systems produce images with more than 8 bit color depth. Converting to another format will in many cases reduce the color depth to 8 bit as many programs and some image formats like JPEG don't support higher color depths.

JPEG is lossy compression, I would always prefer PNG. You might have to use JPEG at some point because of the file size, but that should be at the latest point possible.