How to fix sagging ceiling?
Is it a reasonable approach to resolve the current/future sagging issue?
Yes, adding 2x6 lumber should resolve your issue. 12 feet is still a long span for unsupported 2x6 boards, but it'll be a definite improvement. Be sure to crown each rafter to maximize benefit. 

Could I potentially use a 4x4 instead of a 2x6 to get the same support and save myself the 2" of head room?
This is inadvisable. The additional weight and lower rigidity of 4x4 lumber make it a poor rafter choice. They'd also be quite expensive. 

Am I ok stopping at the wall with the new 2x6 joists?
Yes. 2x4 rafter tails should be adequate. It's not critical that the new rafters bear either on the wall or the ridge beam. Fit them as close as is convenient. The short spans of 2x4 beyond the 2x6 boards have the strength needed to carry the load. You might level off the bottom ends of the 2x6 boards to create a taper into the 2x4, like so (but obviously with less pitch):

                /      /
               /      /
              /      /
             /      /
            /      /
           /______/
          /    /
         /____/


Am I ok to sister the 2x4 and 2x6 joists using screws every 18 inches or so?
Yes, if your screws are 2-1/2" gold construction screws (or better--no black oxide), you use two per 18", spaced widely, and you be sure to clamp the rafters tightly together as you progress.