Just got back at it today and decided to try the easiest tests first. We removed the electric fuel cock and reconnected it to the electrical connector. Then initiated the start with my assistant at the fuel valve with a pair of duckbill pliers ready to rotate it 90* at approx 4 sec into the sequence. We got a normal start with about 275* T-4. and all parameters normal.
The head scratcher here is that with the electric fuel cock connected to the electrical connector-- it rotated 90* as it should and also rotated back 90* at shutdown (I shut it down by turning the Start switch to OFF and using the Fuel Cut-Off lever). The fuel valve is not stiff and is easily turned with the pliers. Is this a characteristic of a "weak" electric fuel cock? It appears that the electric fuel cock is the problem unless I'm missing something else.