Identify/troubleshoot this millivolt furnace valve
Since this question was asked 1 year ago, I have to assume you have solved the problem. If you have not I will give some ideas. first the 360 millivolts  you show should have been 750 to 800 millivolts since this is a pilot generator. You can test the rest of the system by temporarily replacing the generator with a 1.5 volt standard flashlight battery. The condition of the wiring is also suspect; loose connections, old oxidized control contacts, an old thermostat, are all potential problems. I would recommend replacing all the wiring and controls and the pilot generator. As for the gas valve it is probably a "general controls" valve but any 800 millivolt gas valve can be used as a replacement. I recommend you call a heating expert. For the battery test above only connect the battery for a second or two just to see if the gas valve works.