Would the coolant temperature sensor cause the engine to stall at stops and run rough?

Yes. A bad temp sensor can cause a bad mixture in your fuel causing stalling even more so when the car begins to warm up. Replacing the sensor is very easy and very cheap (about 15-20$) and should be done first to eliminate it as a factor for other problems that could be very expensive (Fuel pump which costs about 250-350 depending on your make/model for instance)