You will not hurt the solenoid by connecting it backwards.
All that happens is it pushes the plunger forward rather than pulling ut backwards.
Some do have a diode in them in which case they just work or don't work.
Pull the plug off and check for ground on one of the wires which should not change.
The other wire should + battery voltage when the key is in the on & start positions .
To confirm you have a fuel supply problem
1) put some fuel down the plug hole & crank the engine
IF it fires then the coil is working
2) put some fuel down the carb and crank the engine
If the engine fires then the valves are working and you have a fuel supply problem.
Did you check that the choke was fully closed ?
Kohlers are very sensitive to choke .