If you Google "weather provence", you will find dozens of sites that say what the weather has been like in past years. There is absolutely nowhere that can tell you what it will be like this December.
For driving times, see www.viamichelin.com
NIck is right...you will be hit with a drop odd fee for the rental car of up to several hundred euros. You could take your rental car to a French town close to the Spanish border, like Perpignan, drop the car there and take the train to Barcelona.
Usually weather is rather mild in mid december. It barely snow, but that can happen once in a while (sort of every 5 yrs).
For the time being, weather has been very mild and nice this fall
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