On large aircraft, why do flaperons not run the full length?
This flaperon is actually the high speed aileron. Using the outer ailerons at high speed will twist the wing, which in turn reduces their effectiveness. For every aileron there exists a speed where it becomes ineffective and above that speed its function will even be reversed. See the flexible wing as a flap in its own and the aileron as its Flettner tab of sorts.
By using those high speed ailerons, Boeing can build the wing lighter with less torsional stiffness. In addition, it creates a gap between the Fowler flaps at the spanwise location of the engine exhaust, so the Fowler flaps will not reach into the very turbulent exhaust stream. The 737 uses an exhaust gate for the same purpose.
Using flaps for trim means you need to be able to finely adjust the position of each flap individually. This fine adjustment is better left to the smaller control surfaces or the even smaller trim tabs.