Quoted from Pwedge:

My vote is battery damage on the CPU causing your headaches. Your board has clearly been worked on due to this before, as shown by the socketed U51, missing solder mask in various areas in the vicinity of the battery holder, and tarnished looking component legs in the area of the reset circuitry.
Most likely the alkaline is continuing to leach its way across the copper on your board and into the legs/bodies of the various ICs, which is why you are starting to see the problem get worse. The 6821 PIAs in that area are also responsible for the display segment strobes and the controlled lamps, which explains why you are seeing display segments missing, as well as odd behavior with the feature lamps.