Done it many times:
Myth 1: Christmas Day and couple days around it are the most crowded. Though those are extremely crowded, the heaviest crowds of the year are towards the end of the Christmas week, particularly Dec 29 to 31, with New Year's Eve being the most crowded day of the year when MK and Epcot in the evening are like being stuffed in a small closet with 100 sumo wrestlers.
Myth 2: Crowds drop off significantly beginning January 1. They drop off a little meaning, for example, Soarin goes from being a 3 1/2 hour stand-by line to 3 hours. Huge crowds remain for three to five days depending on when Jan 1 falls and if a Wednesday they won't diminish much before Saturday.
If you go then, do not expect to do every ride and attraction. Christmas decorations won't start to come down until Friday or Saturday and most park ones are likely to still be there Sunday. Everything is decorated and see the outdoor shows and parades which take on Christmas themes, Osborne lights (the most impressive light show on earth), and even Country Bear which switches to a Christmas show. Usually they just have all the parks open early in the morning and going early means you can hit a few things without much problem before the huge crowds begin. Hours also go to late at night and, other than on NYE, that can be a time to get on some things without enormous waits. Fastpass is a must but be aware that for popular rides those can be gone for the entire day and evening by noon. Some things, like Pirates or Haunted Mansion, can have huge lines in the morning but between 4 and 7, they can be much lighter.
If you want to eat at a buffet or table service restaurant during normal dinner hours you must make ressies many months in advance.