A bad coolant temperature sensor will still be bad with a jump start.
Unless the sensor was working intermittently, or the engine was warm enough to be right on the verge of starting anyway.
Typically, with a bad coolant sensor, the colder the engine is, the more difficult it is to start. In moderately cool weather, the engine may still eventually start after a lot of cranking, as it warms up a bit.
A new battery can still take a dump early on, my Pathfinder battery failed at 7 months old.
Yep - Batteries tend to have a high "infant mortality" rate.
However, if a battery is spinning the engine quickly, it ***should*** also be supplying enough voltage for good fuel and spark. Usually, by the time the fuel pump and ignition stops working, the battery is too weak to spin the starter anyway.
Here's the resistances a good coolant temp sensor should be producing, when measured at 68, 122 and 194 degrees F.:
If you've got a DMM, it shouldn't take more than a few minutes to measure the resistance of your sensor to see if it's in the ballpark.