On a rainy day, just make sure you have waterproof shoes and a spare pair of dry socks. Rain coats are great, but I have to say, last month we were there for Thanksgiving with torrential downpours and actually found cheap ponchos worked better. The rain was so bad it soaked through out "waterproof" jackets. Honestly though, most of the time, the rain isn't that bad. Be careful on IASM...we thought it'd be nice to get 10 minutes of dry relaxation, but we failed to account for the fact that the boats would be outside as well. We had easily about 2 inches of rain accumulated in the area where you put your feet. Otherwise, good places to hide if it rains too hard include Animation Academy at DCA and Steamboat Willie, the Lincoln show, and Tiki Room at DL. Try to snag an indoor table for meals, too. If you don't already have reservations, try to get at least one a day so you're guaranteed a dry place to go.
As for crowds, it's a toss up. Usually, crowds are lighter in the rain. But, you're going to be there right at the highest peak time of the year. If there's any chance of rain 12/26-12/31 since it's mostly people who have planned a vacation vs. locals with APs, it will probably still be crowded. But you might have luck in the days leading up to Christmas. That's been my experience at least.