We go every other year Dec. 26 - Jan. 2 and take a high school girls basketball team! Our experience has been that Jan. 1 is a great day to go to the parks - I'm not sure if everyone is sleeping in or leaving that day, but that has been the least busiest day when we've been there.
NYE in EPCOT is great - even trying to keep track of 24 high school girls. The park is the largest but they do 2 fireworks show, one around 9 and one and midnight. So after the first fireworks, people with young kids and strollers leave. The countries of EPCOT also have their own parties based on how each country celebrates. Our plan is that as soon as the last firework goes off at midnight, we all run for Disney transportation.
During this time, there are a lot of sporting events happening at Wide World of Sports (this is why we're there.) There is a boys and girls basketball tournament (last year over 100 teams with their fans), a soccer college preview tournament (it's HUGE), a wrestling tournament (about 40 teams) and for some reason, baseball games!!! The basketball players and wrestlers stay at CSR and some of the All Stars. The soccer players stay everywhere.
Yes, some of the parks will close in mid-afternoon due to crowds. We've been to MK when it has closed and Disney opens up a special walkway (usually used just for cast members) to get from the gate to Fantasyland because Main Street USA is so crowded.
We have been able to swim every year but 1 (it was the coldest winter ever in 2009-10). Last year, the weather was perfect with us just needed jackets at night and it never got really hot during the day.
The parks are beautiful during this time - crowded but beautiful.
We'll be back 2015-2016.