The car will start fine even at -10c with the wire disconnected. The purpose of it wire is to heat the bi-metalic spring in the choke control housing. It starts to bring the choke off before the temperature of the water circulating in the auto choke housing warms the spring. So without it you will have a longer period with the choke fully on and the fast idle.

TT

as tt says
im pritty sure its powerd from a relay switched by an ouput on the hybrid ign module with cvh carbed engines

the alternator works by being excited from the small terminal (warning light) this should be connected from a switched supply (usually ignition) so one end is live from the switch into the warning light and then to the alternator.The other large teminals on the alternator go directly to the battery, the alternator is earthed through the engine itself to the earth strap to the battery. Check your earth conection and correct the wiring it will work, if all is ok with the altenator.Hope this helps!