Dry ice is simply frozen carbon dioxide (the stuff we breathe out) and completely safe as long as we have oxygen to breathe in. The only dangerous part is that it is much COLDER than regular water ice. Therefore you can easily get frostbitten from handling it without gloves. So although it is a chemical (so is water, milk, sugar), it is a safe chemical and is safe next to food.
Great cake! I wish my mom would have made cool cakes like this