I don’t like to leave only a comment so I’ll turn mine into an answer :

Most of the time in that case, I’ll just say “Il fait froid à Paris ?” and the answer will often be about how cold it is and not just if it’s cold or not.

And if you want to be really specific about how, my choice would be “Quel froid il fait/fait-il à Paris ?“. I find the sentence “À quel point fait-il froid…” a little heavy and unnatural.

You can’t say “Combien froid fait-il ?” but you can say “Combien fait-il à Paris ?“, and the answer will be in degrees: “Il fait 3° “

À quel point fait-il froid à Paris en décembre ?

This is the best solution. As @SimonDéchamps said in his comment, you have to invert froid et fait-il from the sentence you wrote.

Il fait froid comment, à Paris en décembre ?

I’d totally say this in an oral conversation but I think when writing it is better to use the first solution.

However Combien froid fait-il à Paris en décembre ? is not a good alternative.