Dear H.,
If you're not comfortable, don't do it. Get a second opinion or look for some other causes of the problem.
Having surgery seems to me like a drastic step to take for a child that isn't sleeping well. The way you described your office visit would have me questioning the validity of the doctor's diagnosis as well. Just because he's a "specialist" doesn't mean you shouldn't question him. He may have the title, but that doesn't necessarily make him good at what he does. It just doesn't sound like he did a very thorough examination, and I would definitely get a second opinion before putting my child through surgery. Although all the responses you received indicate that the children did fairly well with the surgery, it's still surgery and there are significant risks with such a thing. Also, the tonsils and adenoids are lymphoid tissues, which is the bodies primary site of immunological activity against disease. If you take them out, that first response to disease becomes the job of the lymph nodes in the neck, but that's not supposed to be their job so they can become really compromised and lead to other problems for a person.
My daughter, who is 3 1/2, has similar issues to what you described. She has never been able to breathe out of her nose, especially at night. She snores. She wakes up in the night and can't put herself back to sleep without coming to my room (but she has no problem if she gets up to potty). She also was having recurring ear infections. I took her to an allergist and found out she is highly allergic to dust mites and cats (we have 2). I have put dust mite covers on all the bedding, am obsessive with cleaning her room, clean the cats, have a spray to neutralize the dander in the house, and she takes an allergy med once a day. I have also severly limited her dairy intake because dairy can increase the mucous production, and her body was having a hard time dealing with all the mucous and getting it to drain properly. Since taking the steps to deal with the allergy, she's doing so much better, even though we still have the cats. I would look into some other causes of the problem before deciding on surgery.
It's a hard decision. Good luck with it.