When should I pack an image into the file? When would I not want to use the tab "pack an image as embedded data into the .blend file" ? If I do use it should it only be at the end before final export?
Packing images has nothing to do with exporting. Most exporters have their own ways to deal with textures. 

Blender's packing feature allows you to include external resources in your .blend file. This comes in handy when you want to distribute them to other people or showcase them on Sketchfab.