When I first got my B7800 and before I had the chance to check everything out I had a similar problem that you describe. Mine would just stop running and choke to a stop, then start up a few minutes later and run fine until it stopped again. Turned out to be a very dirty fuel filter even though it did not look that bad and the site glass looked good. Apparently the engine would suck fuel faster then the filter could allow, then when stopped the fuel would catch up and run good for a while. New filter and never had that problem again. That being said and you are sure there is no power to the connector I would check the continuity from the connector plug to the start relay behind the dash. You can test the solenoid by applying 12v + from the battery directly to both terminals on the solenoid. If you can hear it click the solenoid is good. If it does not click, run a wire from the battery neg. and touch the body of the solenoid, if it clicks you have a bad ground, if still no clicks you probably have a bad solenoid. Hope this helps at least a start.
Buck