If the cart will start by jumping the micro switch from the gas pedal, you have found your problem and it is not the solenoid. Start by spraying the inside of that switch good with WD-40 or contact cleaner.
Although I find it interesting that you say it will start by jumping the micro switch, which means the switch is bad, and you also say it will start up by jumping the solenoid. If it starts by jumping the solenoid, you are getting contact at the switch. But if it just turns over, when jumping the solenoid, that makes sense.
I prefer to use a light that you can buy at an auto parts store. It has a probe and a wire with an alligator clip. Put the clip on the neg. at the battery. Now you can check the wires at the switch. One side should be hot all the time, the other side hot when you depress the gas. If that checks out, depress the gas and you should have a hot wire at the solenoid. If you have a hot wire there, put a wire from the neg, to the other side of the solenoid, while depressing the pedal. If you get a start, you have a bad ground. If it is not the ground, it is the solenoid. But I still think that from what you say, it is the switch.
Keep us posted.