As I understand it, the sensor that feeds the Temp Gauge does not have anything to do with sending a signal to the PCM to regulate fuel/air mixture. However, the other sensor (ECT) does send a signal to the PCM.
I will volunteer this info on my '98 SOHC...I brought it up from AZ about 5 yrs ago and it seems it's always had similar issues to some degree when the temp falls below +10 degrees or so. Always joked that it's a desert truck and just can't handle the rigors of our Northern Plains cold. It runs flawlessly, otherwise. Since I've owned it, I've replaced the IAC, plugs, wires, t-stat housing w/Motorcraft sensors, cleaned the MAF, changed out the intake manifold gaskets, pcv/hoses, fuel filter. So, with it being -20 degrees Sunday morning, I went out to start the truck after it sat overnight and, not surprisingly, it behaved much like you describe. It started and stalled immediately (I've come to expect that on very cold days). Started it right back up and the idle would surge to maybe 1800 rpms...then fall to where it wanted to stall had I not feathered the throttle back up to 1500 or so rpms. I did this for several cycles over the next 30-45 seconds and finally, out of curiosity, just let it stall. Waited about a minute and started it back up. Ran perfectly fine from then on. Today, it's in the low +30's and, after sitting for about 24 hrs, it started right up and idled perfectly as usual. So...who knows?