I am seriously hoping that the Einstein and the Time Traveling Shoe will be dealt with later. Isn't it an adventure game factoid that when a subject is adressed by the characters more than once, it is going to play out in the plot later? Anyway, I think the Timeline B of 1931 will be choatic. 1986 Doc could have had Einstein with him around the time the first speakeasy blew. If Edna was the culprit and Doc was the only witness, the brick incident could of not been an "accident." If so, then Einstein attacked Edna like a guard dog, grabbed her shoe in the hassle, and then got scared by the fire and ran to the DeLorean. The retrieval feature activated and Einstein and the shoe were sent to 1986. All this could be before Doc's death in Timeline B.
Marty's mission was to break 1986 Doc out of jail with the rocket drill of 1931 Doc. Old Doc was sprung from jail by Kid and the plan changed. The hectic rescue happened (distraction #1) and then Marty fading away (distraction #2). The new mission was to save Arthur from being offed by Kid's gang. This was distraction #3. Three crazy distractions are going to make you forget simple things like retrieving your dog. So, Einstein was left alone in 1931 and sort of adopted by 1931 Doc.
Einstein was Emmett's lab rat and I can see how this could effect Emmett's timeline. For example, it could speed up the pace of Emmett's inventions but it wasn't addressed as a threat in the game. Doc didn't seem worried. At the end of the episode, they were calm and collected enough to remember to take Einstein with them. Yeah, it was dangerous to leave him there, but I doubt it was a factor in the Citizen Brown 1986. Or maybe...