system
Could you explain what you mean by a "backwards connection"? Do you mean having two drain flows coming at each other and meeting together where the line exits to the septic tank?
Similarly, does the "end clean out" refer to the one at the end of run above?
Yes to both of the above. If you use an AAV, it does not have to be extended to a point outside the building. If you do use a AAV, however, since it appears that the drain portion will be connected at the lowest point in the sewer system, if the line to the septic tank floods, for any reason not just because of a stoppage, that sink will not drain even though everything else might operate normally.