My car is making weird noises when starting up in the cold.
A grinding noise when starting car is usually caused by the starter. In this case, since it’s happening for a short period after the car is running, the noise is the starter failing to disengage.
When you start the car, the motor in the starter spins up and a solenoid pushes out the pinion. The pinion has teeth that engage on teeth along the edge of the flywheel or flex plate. This turns over the engine. When you let off the starter, the pinion should go back to its original position. If it doesn’t go all the way back, the teeth can grind against each other. This binding can get worse when it’s cold, since the parts are a little smaller, shrinking clearances.