It's unclear to me what are the maps. In particular, I have a problem with the $0$-th dimension situation. Suppose $r,s\geq 0$, then $F,E,B$ are all connected and thus $H^0$ is free of rank 1, and the beginning of this sequence can't happen with the obvious maps. In n-lab the homological sequence stops in degree 1, is this also the case here?

Yes, thank you. I'll edit something about it.

Another nice source for the details of the connection John Rognes talked about in the form of guided exercises is Weibel's K-Book chapter IV exercises 3.9,3.10,8.5,8.6.