This paper formally states all of the assumptions necessary in the proof of Cox's theorem (R1-R5 in the paper) and notes where the controversies are before going on with the proof. R5 is obviously not well supported and the major dispute over R1 is whether plausibilities must be universally comparable. (R1 and R5 correspond to your two major objections above, in order).

As requested below, a top level post would be very interesting