Receiving a photo via iMessage: when is it downloaded? I am downloading a photo over iMessage.  When I see the previewed thumbnail image, is the entire photo downloaded? If not, when I open the photo will it be completely downloaded to my phone? Like stored in a folder of some kind? At what point will it be downloaded completely?

Also, if I save this to camera roll, does it transfer folders?

Just a theoretical question; no real utility from it.
I don't know about the theoretical, but the actual choices that offered in Messages include tapping twice on an image to access the Copy command, or tapping once to open in the image in its own window, in which case the usual arrow in a box appears, leading to a number of saving and sharing options, depending on what you have installed.

On the Mac, messages and photos (or any other attachment) are automatically synced between devices and stored in ~/Library/Messages/Archives and ~/Library/Messages/Attachments.