Small Engine Troubleshooting Problems? Ask a Mechanic Now.
This sounds like you have a broken or loose wire from the start switch, or possible a miswiring of a wire.
Yes, it is very possible that all 3 switches are acting up - especially if the harness itself is the problem.
As far as where something could be miswired, I have no idea what all you have done to the unit. You've already said you removed all the wires from the solenoid, and it is very possible you have something wrong there.