When I see the previewed thumbnail image, is the entire photo downloaded?
Yes and no. Obviously, to be able to see the entire thumbnail image, the entire image is downloaded, but it is compressed. Think of when you save a jpeg with different quality settings. To save processing, the thumbnail image is displayed in a lower quality, that is why the loading wheel spins when you tap on the image to view full size.
If not, when I open the photo will it be completely downloaded to my phone?
I'm not 100% sure on this one, but my guess would be that it is only stored in RAM at that time. It would make no sense to store it on the file system and still provide an option to save to the camera roll directory and then have to remove it from the temporary directory.
At what point will it be downloaded completely (if ever)?
Once you save it to your camera roll, it is downloaded fully. You can confirm this by turning your phone off and on again, the photo will still be there, which proves it is now stored on your drive.
Also, if I save this to camera roll, does it transfer folders?
This goes back to my answer of the second question, "If not, when I open the photo will it be completely downloaded to my phone?". If my guess is correct, then the photo is transferred to the camera roll directory when you save the photo.