Question about the coolant temperature sensor

Hello municipality, my dicker is not running around again at the cold start. I have the whole procedure spark plugs, plug contacts and vacuum system behind me. The coolant sensor was changed once on suspicion, but then the old one had not changed. However, there was still unknown the NWS in the bucket. Just to be sure: can the coolant sensor be responsible for that the motor “hops & shakes” during the cold start during the ride the Moto r just let him kneel down as if he was choking off? No fault code is stored and the FOH is on holiday (to which I am not going anyway) Does the coolant sensor leave a mistake at all? Or does it only slowly break down and only then when he is really in the bucket does this report itself? How last week the engine went off and the symbol with car and fork key lit. No NOTLAUF after the engine restart was the lamp out and also no error in the memory. elesen via gas and brake pedal… HELP I just put down 500eur so that the planer runs again and now again what… What does the exchange of the coolant sensor cost? Thanks for your time and help