Hello chemicalX,
This depends mainly with the agreement you have with the developers you are collaborating with. If they say you should be the one in charge of exporting then after you update the file you should probably also export with the settings they have given you.
Typically asking the animator/rigger to export is uncommon, and generally it is enough to sync the Spine file and the new images file so that the developers can export with their preferred settings on need.
I think the best way to get precise answers in this case is to ask your point of contact what is expected of you. 😃
To answer your organization question, when I have to add new skins to a project I may be sent the new files, and I'll typically merge them in the main folder of the project, discarding the separate files once I'm done integrating them.
