1) Starters do nothing but start, so changing the starter would have nothing to do with the motor stopping.

How cold is cold? If we're not talking below freezing, is it possible that it misbehaves when it's damp or raining? In an older vehicle, seals can be decayed and moisture in a cold engine can short circuit stuff.

Motor stops when it doesn't get spark (electricity) or fuel.
Fuel stops when the fuel pump stops. The fuel pump stops when it doesn't get electricity (assuming that it's a modern fuel pump which is powered by electricty. A new alternator is a good idea. So is a new battery.

Sharp rapping from the vicinity of the glove box. Let the little guy out! It could be that a belt is slapping while stiff and cold, but those are hard to figure without direct ears to the noise.