Thanks! Let us know if you have any questions about the proof itself--unfortunately, it's not particularly well-documented, and Coq isn't super easy to read in the first place.
At this point, the standard intro text is Software Foundations [1]. I highly recommend it; it will teach you Coq and also probably make you a better programmer. After SF, Certified Programming with Dependent Types [2] gets more into the practice of proving serious programs correct. These books are both available online in the form of literate Coq files.
As far as online lectures, OPLSS [3] often has Coq lectures which are quite good.
One could imagine doing some kind of differential testing of another key-value store against vard to ensure that for a random but large number of requests, they return the same value. I'm not sure that doing that would gain you anything over using Jepsen [1], though. One could also imagine implementing another key-value store in Verdi and proving it equivalent to vard; reasons to do this might include using better data structures or serialization mechanisms, or a different consensus protocol. I'm not sure that this would require significantly less effort than proving this new key-value store (usrd? whittend? :)) correct from scratch, but I'd love to see someone do this!