Yep, That's the speed sensor wires. Looks like your speed sensor is broken off. You'll need a new one. $96 on Partzilla or find a good used one on ebay for about 1/3 that price.
No, plunger should just pull out of the carb and stay on the end of the cable. You may have to use needle nose pliers to twist/turn it out of its bore if its stuck. May need use a little penetrating oil on it and let soak if stuck bad.
Some of the Chinese clutches work ok but not sure if they're rebuildable.
I rebuild clutches for around $250 or will buy yours if its not too far gone and is rebuildable
I know this is a old thread but jump 12v directly to the starter terminal to check starter function. If good, then jump solenoid terminals. If good, you probably have a wire connections issue or starter button problem.