The year we went from Dec 26 to Jan 2, most parks were full to capacity every day. It was amazing, hearing about it! We had a great time, but really enjoyed our time at the resort (AKL). What made it really nice and comfortable was that we were on concierge level. This was nice for two reasons: they made all our ADR's, activity plans, and such for us; and we really spent more time in the lounge than many would during a slower time of the year.
We went to a park fairly early in the day, enjoyed it until the crowds got very heavy, then we'd go back to the resort, relax, the kids would swim (run from pool to hot tub and back!). Then have dinner somewhere. As we approached NYE, the crowds became noticeably heavier earlier, but the early times were fine. Now, NYE we spent maybe an hour or two at Epcot, and I really couldn't stand it - got a bit crabby with DH

The concierge staff had booked us dinner at Flying Fish, sad to say it's the second worst meal we've had at WDW, but hey that happens, then an Illuminations cruise on Breathless (we didn't even know what that was, I just had asked if sometime during the week we could get an Illuminations cruise). It was so cool to watch the show and fireworks while the folks above us were literally smashed in...
I would definitely stick to lunch ADR's if it were me. The evenings may be very crowded.
And to specifically answer one question: from our experience, Epcot was closed before 3pm on 12/31 - to capacity. I wouldn't have wanted to be there if I could. The atmosphere around the Boardwalk and the Yacht and Beach Club was very festive. I don't know what the current economy will do to affect this situation, but you will probably have people living in the area who will go to celebrate, so I'll bet it will be full. I wouldn't plan dinner that night in Epcot, if it were me.