I work as GM of a fine dining restaurant. We are a farm to table restaurant that caters to many diners with allergies. FIRST OFF. Do not fret. This is becoming more and more common Disney is especially well equipped and will have answers for you on what you can and cannot have. As time goes by even smaller restaurants will become more and more accustomed to it, as it seems that more and more people are eating gluten free. Disney most likely has celiac printed menus for you but I'm not 100 percent sure on that. As far as dealing with your allergies in a less corporate situation. I have some advice....
If you are eating at a fine establishment that understands gluten allergies I always recommend picking out a dish you would like, and then allowing them to offer what they would do to tailor it to you. This is much easier than saying "what can I have on this menu". Because honestly you can have many things. I think it's much more helpful to have a guest say. "I am interested in the Snapper dish, but can that be made gluten free? Are there any hidden gluten allergies"
Many restaurants will offer you something in place of bread. But some will not. Please do not demand something for free. (You are clearly far too considerate for this, but I have had guests try to demand fresh vegetables, as a replacement for bread, for no charge).
Start to understand the difference between "may contain, and cross contamination issues"
For example... The grits we serve as a side dish are indeed gluten free. But we do not process them and they come from a place that does process items with gluten. So although it doesn't contain it, it may contain trace amounts that depending on the level of severity of your allergy could aggravate you.
Finally, be nice. Service professionals won't you to have a wonderful worry free meal. And being nice will go a looong way.
You have nothing to worry about. You are coming from a place of consideration. And as for Disney, they are over prepared for you.