== Special relativity == 
 Your comments at Talk:Special relativity#How does it work? indicate that you are assuming things about special relativity which are not true. All your questions are loaded questions which explains why we are having such a hard time answering them. Special relativity is much more similar to classical physics than you are giving it credit for being. 

 Perhaps you were confused by the fact that the usual derivation of the formulas of special relativity makes use of simplifying assumptions, to wit, that both observers have the same origin for their reference frames and that the axes of their frames are parallel and that the relative motion is in the x-direction. These simplifying assumptions are just there for teaching purposes and are not in the theory itself. 

 "... the relative motion between two observers is the key explanation of their difference of appreciation of their respective distance to a material target (e.g. a firecracker)." 
 No. You might have gotten this mistaken impression due to the simplifying assumption that the origins are the same. 

 Measurements by the observers are supposed to be independent of each other. 

 There is no reason why any criterion is needed to determine "which observer will find the larger value". As I said before, the result depends on the factual situation of the observers. 

 If "two observers [are] moving away from the firecracker in opposite directions at ... the same constant speed v/2" having started together with their origins at the firecracker, then indeed they will measure the distance to the firecracker when it explodes to be the same. You need to be careful about specifying the time here since in special relativity the simultaneity of two events (not at the same location) depends on the observer.