While a cold battery has less cranking amps and total capacity available, it also has a significantly slower rate of self discharge.
If fully charged, it will not freeze until - 78f or somewhere around there. it depends on the actual specific gravity of the electrolyte which might be higher in batteries intended for colder climates.
Newer healthier batteries self discharge less than older less healthy batteries.
The parasitic draw might be the same whether cold or warm, but a warm battery has more capacity, and cranking amps available.
As far as the alternator, it is controlled by a voltage regulator. The higher the voltage the VR chooses, the more amperage the depleted battery will ask for.
In my own vehicle, fully fielding the alternator, in 65f ambient, at idle, has it approaching 220f in about 15 minutes, and 220f is about as hot as one wants it to get.
How hot the alternator gets is related to how stressed it is.
How hot an alternator will get in another vehicle is an unknown with tons of influencing variables, one of which is the fan rpm of the alternator itself and the radiator fan's flow around the alternator in that specific vehicle.
The battery decides how much amperage it will accept at a certain pressure( voltage), upto the amperage limits of the charging source.
This amount varies with depth of discharge, battery size, health and temperature.
A large well depleted healthy battery, in a vehicle whose voltage regulator decides to hold voltage in the mid to high 14's for 15+ minutes, might put the alternator up in the 220f range, at idle.
But few vehicles allow this voltage for such durations, even if the battery is well discharged and would benefit from it.
Low temperature long term battery charging/maintaining, ideally, has a charger with a temperature sensor, which will raise the maintenance/float voltage.
My dad bought a Noco Genius 1 for ~ 25$. It has a temp sensor inside the charger itself.
At 75F , I observed its behavior with voltmeter and ammeter.
It is a 1 amp charger until battery voltage reaches 13.65 or so, then a 0.5 amp charger until battery voltage reaches ~13.9, then a 0.2 amp charger until battery voltage reaches 14.7v, then it shuts off.
It kicks back on with 0.2 amps available once battery voltage dropped ( from parasitic loads and or self discharge) to 12.69v, and will stay on until battery voltage reaches 14.1, at which point it shuts off again and waits until battery voltage drops to 12.69.
In my opinion this will get a healthy battery to fully charged or very nearly so, and allow it to self discharge to ~99.5% before kicking back on.
More than good enough for most vehicles.
These observations of the NOCO genius 1 algorithm, were made with a small AGM battery and the Noco in AGM mode. Kind of an accellerated test with an artificial load applied, reducing battery voltage to that 12.69v. Behavior on a larger more discharged battery could be different.
