No, it's not possible to "scan" any fan edit, since none of them are done on film. As for ANH:R, it was done at DVD resolution, so it's not even possible to make that HD. He didn't use an HD broadcast for ANH, he just used the official DVD.

ESB will be 720p, from the German HD broadcast, not 1080p because it's just too big to work with and some effects back when he was attempting 1080p didn't turn out so great because the increased resolution limited what he could use (the wampa tusk test comes to mind).

2K is slightly better than 1080p resolution, and 4k is significantly better resolution than 2k.

SD (NTSC): 720x480 (width first, height second, and this uses non-square pixels - with square pixels, a true 4:3 image would be 640x480)
720p: 1280x720
1080p: 1920x1080
2K: 2048×1556 (full-aperture)
4K: 4096x3112 (full-aperture)

The Dark Knight had its 35mm scenes scanned at 4K and its IMAX (70mm) scenes scanned at 8K for its online edit. Some other films have been scanned at 8K for Blu-Ray release (Baraka, The Wizard of Oz), though I imagine that's mainly for archival purposes and "future-proofing" than anything, since Blu-Ray tops off at 1080p.

But things that are done digitally are limited to the highest resolution that was worked with. If I'm not mistaken, AOTC and ROTS were shot with 1080p digital cameras, so they'll never be able to look better than Blu-Ray. Though I could be mistaken, as most/all digitally-shot films today are at least shot at 2K, and I don't think George is *that* big of an idiot.