That is correct....an ADR won't get you in once the park is closed due to capacity crowds. Furthermore, if you have an ADR that you had to give a credit card number for (like Cindy's breakfast), they will still charge your card for not showing up! At least it used to be this way. So the planning and responsibility is totally up to you.
I don't know about whether you get a priority when people leave....I would guess it would be whomever is waiting in line next to get in.
Dec 27-30th will be very busy....but probably won't reach capacity. Dec 31st most likely WILL reach capacity - probably by mid-morning. If you want to stay in MK, you will have to STAY in MK - all day.
What the previous posters have said is correct. WDW has several "levels" of full. I think the first level of full has something to do with the parking lots, then it moves to no more off-site guests admitted, then absolutely no one else is admitted (and Goofyfreak is right, your family being in there doesn't help you any). On-Site Guests are guaranteed admittance to "A Park" just as Avalance said. If you want to go in MK and it is full, they will happily transport you by bus to AK, for example, and that satisfies their guarantee.