can anyone confirm my understanding of the basic physics here?
they are observing a transition from one particle to another that involves a process whose probability depends on "all particles" in some weird sense, because it includes particle-antiparticle pairs (so is affected by vacuum fluctuations?).
because of this, they have a weird situation where an experimental result depends on all possible particles (below some energy cutoff i guess).
and that is why it's a good test for standard model v super-symmetry. because super-symmetry (to be symmetric) predicts a bunch more particles. and if those are real, we would see something different.
i am not sure where i got this from, but it seemed to be implied by one of the links here...
Yep, you've pretty much got it right. It is an indirect probe of new physics, not a direct probe. However, that method is well established. Some of the strongest constraints for new physics come from things like neutron dipole moment and studies of the g-factor of the muon (see g-2 or "g minus 2" online). These indirect methods make use of the virtual particles that arise from vacuum fluctuations in ways that are cleanly calculable (and thus make quantitative predictions).
All that said, SUSY has about 100 free parameters in the model, and all of these LHC searches hoped to find SUSY in the easiest of places, even though there is no firm prediction from the theory. Simply put, for a long time to come, SUSY can be "tuned" to remain consistent with all existing physical data.