You're definitely correct that getting sychrotron time is hard :(
On the first part though: Yes and no. XRD will tell you about things like strain and unit cell size distribution, so in that sense, you can't resolve a single doping site. On the other hand, if you have a reaction going on, or some dopant diffusing into your sample, synchrotron XRD is powerful/fast enough that you can "film" how the crystal structure changes in real time. That "film" will be a kind of average of many sites, but can still be focused to a relatively small region (don't remember exactly how small off the top of my head, but I believe we're talking nm-scale).
Not 100% comparable, but synchrotron XRD allows for real-imaging of solid state chemical reactions and can, in a sense, resolve the unit cell structure of the crystal. However, what you get from an XRD is nothing like this "photo-like" image, but a diffractogram. I think you could probably re-create an image like this from a 2D diffractogram though, but I'm not sure.
I’m assuming they’re indicating that the mass below the apparatus increased in fall (when storage was filled) and decreased slowly through the winter, leading them to measure a changed graviational constant. A back of the napkin calculation shows that in order to change the measured gravitational constant by 1 %, by placing a point mass 1 m below the apparatus, that point mass would need to be about 15 000 tons. That’s not a huge number, and it’s not unlikely that their measuring equipment could measure the gravitational acceleration to much better precision than 1 %, I still think it sounds a bit unlikely.
Remember: If we place the point mass (or equivalently, centre of mass of the coal heap) 2 m below the apparatus instead of 1 m, we need 60 000 tons to get the same effect (because gravitational force scales as inverse distance squared). To me this sounds like a fun “wandering story”, that without being impossible definitely sounds unlikely.
For reference: The coal consumption of Luxembourg in 2016 was roughly 90 000 tons. Coal has a density of roughly 1500 kg / m3, so 15 000 tons of coal is about 10 000 m3, or a 21.5 m x 21.5 m x 21.5 m cube, or about four olympic swimming pools.
Edit: The above density calculations use the density of coal, not the (significantly lower) density of a coal heap, which contains a lot of air in-between the coal lumps. My guess on the density of a coal heap is in the range of ≈ 1000 kg / m3 (equivalent to guessing that a coal heap has a void fraction of ≈ 1 / 3.)