As far as I know there is no way to prime the pump in these corollas as it only primes after car starts cranking. In my honda city 2017 I hear the pump priming when they key is turned to ignition position without cranking. On this corolla never heard any pump priming noises when key is on ignition. Pressing accelerator without cranking does not seem to make any priming noises either. Once car had separate starting issues which I got fixed. Did get the pump checked with the usual method of taking out the fuel line and when engine was cranked, fuel was pouring out easily. But, the engine temps were warm at the time and not dead cold. I don't really think the temps should be affecting the pump's pumping abilities though.
As regards to the temperature, if the thermostat was stuck open I should be hearing the fan constantly being on but I haven't, but not totally sure of that as have not tried to specifically hear for the fan noise. Will get that checked out too. I basically see temps dropping after I coast to a stop. For better fuel economy I always try coasting to a stop, which in this weather should cool temps down cause no fuel burning inside the engine when coasting in gear and cold air running through the radiator at high speed.
