First and foremost, a 2-year-old does not need a big birthday party. They just wind up hyped up on sugar, stressed out and melting down. Keep it simple, and keep it down to a couple hours.
My daughter is born on Christmas Day, which makes things doubly hard for us. She is turning 5 this year, so we are still trying to get into the swing of what to do for her. We had her first 1/2 b-day party this year, and she LOVED it. Having a half-birthday doesn't mean that you ignore their actual birthday - we had the grandparents over on Christmas Day for dinner and birthday cake, and we sang to her and gave her our presents. Then this July she got to have a Luau party with all her friends. Some people brought gifts, and some didn't. We did not give her gifts, since she got ours on her birthday. It was SO much less stressful for us, and she had a blast.
As far as the "no gifts," some people prefer that just to keep clutter to a minimum. However, if you only motivation is to increase attendance, I wouldn't put it there. One thing we do to control clutter is to have a de-stashing before any gift-giving holiday. I just tell my girls that there isn't any room for new toys, and if they want presents, they have to get rid of something old. They have never made a big deal of it.
As far as themes, you can do anything you want. It doesn't have to be a Christmas party. In fact, I would stay away from the Christmas theme. This is supposed to be a *birthday* party. Do whatever your daughter likes - horses, dolls, tea party, pizza party, a local inflatable bounce place, an indoor city pool, or anything else that strikes your fancy. You don't have to make it into a huge thing if you don't want to.
Best of luck,