I was bored.
Yes, I was referring to the idler pulley on the power steering belt.
As for ice in the coolant, there is a pipe from the heater core to the rear of the water pump so... is your antifreeze in good shape? Is the temperature slider on the heater controls to the hot or cold side when you start it?
The starter theory was just a stab in the dark thinking the cold gear may be hanging up when you start it, then vibrating back in.