N.,
First off the wheat-free, dairy-free diet can help if you think your son in intolerant or allergic to these foods. You might want to talk with an allergist who specializes in children. There are a lot of great options at the health food store. My son also has food intolerances, we use spelt bread (it's another type of wheat, but one that is not used very often), it's expensive, but I use it sparingly. Also rice cakes are a great option. Rice milk or soy milk can be a good substitute for milk. Talk to your ped. about his calcium intake and how to ensure he is getting enough if you eliminate dairy.
Just remember, if you are baking brownies or other sweets from scratch you will have to add a bit of fat, like oil, to make up for some of the fat in butter and milk. Check out Alton Brown's new book, "I'm Just Here for More Food". It's a great book specifically about baking and what each ingredient does.
You might want to consider taking your son off food dyes. Some kids react badly to the dyes that are in kool-aid, juice, snacks, processed food, etc. Try going organic, there are fewer dyes and most of them are natural rather than processed.
As to time at the dinner table...well, he's three. His attention span is not going to be all that long. With our son we offered snacks twice a day (before lunch and before dinner), but told him that he would not get anything until the set snack time...or the next meal. By enforcing meal times and including him in conversation, dinners slowly improved. Our son is 6 now, and his manners are improving.
You said your son has asbergers, you might want to try laminated picture cards of food that he can have and post them on the refrigerator. I heard of another mother doing this with her child. She made up a number of cards based on nutrition guidelines. He could only have one sweet a day so there was only one sweet card on the fridge. When it was used and he was hungry he had to pick from the other cards: carrots, celery, grapes, oranges, apples, etc. The cards were placed in an envelope on the counter as he used them. This helped the mom ensure her son was getting a balanced diet nutritionally and approved snacks were available for him to choose from every day. It also gave her child a bit of responsibility, when he would use the sweet snack and what snacks he would choose from throughout the day.
I know that your son is young, but it might help him to remember what he has had throughout the day and what he can have without constantly asking you every hour.
Check out Chris Johnson's book "Meal Patterning" he includes a "food target" that easily shows which foods are really good for you and which foods are really bad. It is better than the food pyramid and easier for children to understand.
Just remember, God's foods (natural, fresh, etc) are best for his growing body.
Good luck!