Which leads me to a question...what happens if it rains?

Does the parade get cancelled? What rides generally close down?
First, it probably won't rain. If it does rain, it probably will only be for a day or two, so if your trip is longer than that, you will probably still have rain-free days and can plan accordingly.
We went in mid-December 2009 when it was sunny the first day or two, then rainy-ish Friday (started raining hard in the late evening) and rainy all day Saturday. If it rains hard, the parks really clear out...DCA was an absolute ghost town Saturday night and the wait for TSMM was about as long as it took to walk the queue. Alice in Wonderland and the Teacups will definitely close in the rain...I'm not sure any close unless it's raining hard. I've heard they will run rides like It's a Small World until the rainwater starts filling the boats! I've heard some of the outdoor rides like Big Thunder Mountain are uncomfortable in pelting rain (I never had the courage to try them). And if it's raining and cold, people will actually go into Innoventions. LOL. I think Fantasmic! only ran for the first show on Friday that weekend.
It gets slippery in the rain.... If I had it to do again, I would buy some sort of rain boots if it looked like it would rain a couple days because the only thing I didn't like was walking around with wet shoes.
Eating can be a challenge because a lot of seating is outdoors...we had a lot of difficulty eating at Hungry Bear because everyone had to sit downstairs to avoid the rain, and people were holding tables for their party when their party took literally 15-20 minutes to arrive. (When we finally sat down, we had finished our meal by the time the person who wouldn't let us sit at her table's family came down.) Eating was really the only thing I didn't like about the rain. If you got something you could eat while standing under a rooftop (corn dog/etc.), then I think you'd be fine.
It is possible they will close the parks a few hours early if it rains--they did that at DL in 2010, I believe, but not during our trip in 2009. Crowds will be reduced, though, so you should still get a lot done.
I had read here that they will run the parade unless it is very rainy, so we didn't make a special effort to see the parade before the rain set in. That was a mistake...I'm not sure the parade ran at all on Friday, even though it certainly wasn't raining hard during the first parade time. We didn't even see the cavalcade--I'm not absolutely certain it went through, but it could have.
But, even though it rained a lot during our trip, we managed to see my priority (Tiana's Showboat Jubilee which ran that winter only) 3 times (it ran a lot more often than the parade). So we COULD have seen the parade if we'd known it would not run in the rain.