– When I turned the key, all I heard were a series of clicks. I did confirm that some of the interior electrics (e.g. dome light) were functioning, although at noticeably reduced levels.
This is indicative of a lower voltage condition.
Suggest you test the battery voltage at rest and again during cranking.
Question 1: How likely is it that this is a battery problem (and not a starter problem, for example)?
You will need to test the battery in order to find out.
Question 2: Does the ability of a battery to start the car depend only on its temperature at the time of attempted start? Or is also a function of the amount of time since the last start.
There are a few factors to consider:
1 Any excessive parasitic draw.
2 Colder temps will reduce the output of the battery, how much depends on the condition of the battery.
3 How well your vehicle charging system can recharge the battery.
4 Condition of the battery cables.
Question 3: It will get up to zero Celsius in a couple of days. Do you think there is a chance the car will start at this warmer temperature without a boost?
Unknown until a load and voltage test is performed.
Question 4: Assuming the battery is the problem, does it not seem odd that a new battery would go so dead in such relatively warm conditions?
Just because a part is new doesn’t always mean it is good.