I think for some Celiacs the skin to skin thing doesn't bother them, but my son is only about 4 years out from consuming gluten and is still very, very sensitive. We avoid him touching things that have touched wheat.
They do washing up after snack, but we requested that he be allowed to use a paper plate for his snack and avoid the washing up. He does wipe the tables down after snack and lunch, but I don't worry too much about that. Now, if your daughter has an anaphylactic reaction to something, I would ask the teachers to even have her avoid washing up at all. She can do her practical life activity at home, perhaps. You can reinforce that work in a safe space for her.
Asher also has other true allergies--egg, oranges, pumpkin, etc--but we don't really care so much about those because it's not as serious for him as the wheat.
You're totally right that sometimes the products look OK, but they are really not at all. Yogurt is a good example of that with Ash. Some have modified food starch in them, which doesn't say "wheat" or "gluten" but it does contain it. He can read labels like a pro and knows what to look for. Although you can't use your daughter as a sentinel now, you will eventually be able to.
His teachers and even the parents and students have been so great about avoiding contaminating him. Some parents will only send gluten free snacks and two of Asher's friends refuse to bring things with gluten in them. One even claims to have Celiac too, which is cute and sweet in a way.
He's in the Lower Elementary class (6-9) but we didn't have any issue with him in the Primary (3-6) either. I think it's just awareness, education and unceasing vigilance.