To answer the table question, the tables seat eight - four along a long edge and two on each end. There is a 12" lip where your plate goes and the rest of the table is the grill. The chef stands on the other side of the grill and cooks your food. The grill is less hot near the edges, but I personally, would be uncomfortable with a small child near the hot grill (burned my own fingers last time). Children are seated on the very ends, where the grill is coolest, although that means it may be difficult to help your children.
We ate here in October and our table mates had gluten allergies. Soy and the sauces contain gluten, as do the noodles they put in. Everything for them was prepped first, with special gluten free soy sauce and no noodles. Mushrooms were cooked seperately as not everyone like mushrooms, and mushrooms only went on the plates of the fungus eaters. The sauces the chef puts on are light, and you are given dipping sauces to use or not use.
They did not let the table across from us order the sushi as an appitizer on the dining plan (there was a loud discussion regarding this).
However, of all the restaurants I've been to in Epcot, Teppanyaki is the one where the staff speaks the least English. We've had curtious service, but it hasn't always been possible to make ourselves understood. So I think special requests need to be communicated clearly.
Edited to add: I like the restaurant a lot and we like going here, but I find its a difficult restaurant to carry on a conversation at. The table setup means no one is sitting across from you. And the tables seat eight, so if your party is larger than that, you'll be split up. For that reason, I think its a poor choice for a large party, regardless of pickiness.