M.A.
Hi! You can make a cake without eggs by just using what's called an 'egg replacer' - you can find it in the health food section of your grocer. But even that isn't necessary. Here's a trick I've used with success: put an extra tsp. of baking soda and an extra 1/2 or 1 tsp of baking powder in the batter. That's it! Eggs are only for 'binding' ingredients together when it comes to cakes. Other people have used a few tsps of ground flax too.
As for activities: Keep it simple! 2 year olds have very short attention spans. I would keep the party no more than 1 1/2 to 2 hours long. You can do a theme regarding fall, thanksgiving, or something special about her personality or name meaning. Then organize things around that theme. A good rundown for the party would be to have an activity such as a VERY simple coloring page or craft according to your theme when guests arrive. Go straight into cake. After cleaning up from that go straight into gift opening. Then do one of those pinatas that the kids pull the strings. Have only something like fruit juice sweetened fruit leathers for the 'goodies' inside the pinata. At that point a 2 year old will be saturated... let them play a game of bean bag toss or something until its time to go home.