It's been said before, but a green horse and a green owner is not a good match.
Another bad match is a green horse and a young, green rider.
When she was 3 years old, my friend's grandmother put her on her barely trained 2 year old Arabian. She was leading her around the pasture, when something spooked the horse and my friend fell off and broke her back. Now, 20 years later she still walks hunched over and everyday is painful for her.
My point is that it's cute and idealistic for a kid to grow up with a horse, it's dangerous. Horses are unpredictable, and the problem is compounded when you yourself don't know much about horses. Please get your children an older horse that's dead broke. Your horse may be super sweet and kind, but it's young, and all the love in world doesn't make it safe.
You and your children will enjoy a broke horse much much much more than you will with a 2 year old. Don't let attachment stop you from getting rid of her either. She will become attached to the next person who brings her food, and you will become just as attached to the next one, probably more so since you will be able to enjoy it without worrying.