My problem was similar to yours and it turned out to be my battery that was only 1.5 months old. My bad battery showed 12.5 volts, but when I hit the starter button, it dropped to 4 or 5 volts. My starter wouldn't even engage at all.
The place I bought the battery from said the internal welds were bad although it read 12.5 volts. Try a new battery.
FWIW, I tried to jump the solenoid as well and mine sparked a lot. Freaked me out, too.