Sorry, I'll be blunt and honest here, but a dog is not a miracle worker. A dog is not a replacement for therapy, nor any other coping tools and strategies that your child would learn.

You're going about this backwards. Instead of thinking 'What can a service dog do for my kid" think "What can my kid not do independantly, for his age range and maturity, that a service dog could help with?" The short answer is that you're his parent, and a dog is just like another toddler running around. Even well trained ones. Service dogs are also used for increased independence- no three year old will be independant, disability or not.

Why cant you tether him to yourself if you fear him running out in public? They make toddler harnesses that are just for that purpose- to keep kids from wandering. It's much safer to have him attached to YOU as an adult, who has 100% control over where you walk, compared to a dog. A dog can bolt suddenly out of fear, can run into the street and drag the child along with him. Even well trained dogs can get spooked or break training.

http://www.animalsandsociety.org/assets/...aviora.pdf
Refer to page 50 - child got dragged when the dog broke a stay to chase another dog.

Page 51 - Children in meltdown often direct the agression at the dog because it is a closer target. .
Some of the more agressive children would hit or thrash out at the dog, startling the dog and causing the dog to move away from the source of physical abuse.
Fortunately the dogs developed a learned sense of when to move in to distract or comfort the child and when to move away to avoid the childs to avoid the child 's anger.

A dog simply providing sensory input isn't going to teach him how to handle the need for input. He needs occupational therapy to learn these things.

A nice pet will do great in teaching him how to be kind around animals, give lots of love and affection, but there would be no need for a public access service dog for a child his age.

Melatonin is a great natural remedy for sleep issues, even safe with children. Anxiety can be dealt with through therapy. He needs to learn to cope with these things before a service dog would ever be helpful.

I recommend http://www.servicedogcentral.org/content/ for more information