If you select a portion of the image, then copy and past into a blank document, you will lose the exif data if working with something other than the RAW (such as jpg). When you crop via copy and paste, you are only copying a portion of the image, not the file, so you will lose the exif.
1. No 2. A crop should absolutely no effect on the data. If you are using Photoshop, do not use Save for Web option - that DOES strip the EXIF. Do a plain Save As - Jpeg at max file quality.
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