Jan 1 and Jan 2 will be hugely crowded days. Jan 1 will still have close to Christmas time hours (MK to 11, MGM to 10 or 11) and Jan 2 may see the same or drop somewhat (in 2005, despite what that site provided above shows, MK and MGM both went very late on Jan 2--MK went to 11 and MGM went to 10 and then had extra magic hours until 1). On Jan 3, MK and MGM hours will fall to the 8 p.m. closing time (Epcot is always 9, and AK will fall from about 7 to 6 or 5). Crowds will drop significantly on the 3rd and then more on the 4th and remain at a steady pace thereafter with moderately heavy crowds (popular rides, like the mountains, can still accumulate 45 minute to an hour lines but that does not exist all day; most other things have fairly short lines)
Weather varies to extremes. They publish an average temperature (something like high 60's) but that time of year is seldom "average." In 2005, it was high 70's to mid 80's the entire week and there was almost no rain all week, and people flocked the pools. Evenings were very pleasant -- light jacket was needed some of the nights. When we did it in 2003, temperature was in the low 60's daily and a couple days it was only in the mid-50's and nights were cold (low 40's and down into the 30's) and virtually no one did the pools.
To determine likely temperature, go and look at what is happening in Chicago, Illinois about 3 to 4 days after Christmas. If there is a huge cold front moving into Chicago out of the artic at that time, then it will be cold at WDW about 3 days later because those fronts move right on down to the southeast and finish off in Florida. If it is in the high 20's to 30's in Chicago and no artic cold front moving through, then WDW will likely be warm.