Q: Is it possible to do everything or mostly everything in 4 days?
It depends on the age of your kids, how busy the Parks are, and what you mean by everything. There are four Parks: Magic Kingdom, Epcot, Animal Kingdom, and Hollywood Studios.
If you don't do table service restaurants every day, focus on just the attractions in the Parks and assuming the Parks aren't closing early for a special event, I can see a plan where you would be satisfied after 4 days.
Magic Kingdom is the largest park. You need at least a day or a day and half there to cover everything. You could likely do a half day at Animal Kingdom and hit the highlights. Epcot you could do in a day. Then a full day at Hollywood Studios.