Overheating

That's what the fan is supposed to do, it needs to turn fan to cool it down when idling. The reason temperature drops when ur driving, it's because cool air blowing through radiator, taking place of ur cooling fan. U need to install original fan back on

I was told it could slso be water pump or thermostat. Could that be correct also?

Well here is how the cooling system wirks. The engine runs, thermostat is closed, this forces the engine to start heating up because thermostat has not reached set temperature. Once engine or thermistat temperature has reached operating or thermostat preset temperature, thermostat opens and water begins to cool down. This proccess is being done over and over. There's an engine temperature sensor that monitors engine temperature, when engine temperature gets to a certain point such as 220* the fans turn on to cool engine down to whatever thermostat setpoint is. So if the temperature exceeds 220 and fans don't kick in, it's not that pump is not pumping. It's water getting too hot and fans are not coming on to cool it down. To answer ur question, no. Fans should kick in