First off, I'm not a fan of Pediatricians prescribing medications for anything beyond routine childhood illness; it is beyond their scope of expertise. However, I do know that is how so many of us begin this road to help our children.
I agree with the neuropsychologist testing. I had it done with Son when he was 10 and it was a grueling three days (done over several weeks). For him, and for me though, it left as many questions as it did answer. It's interpreting the results and how they are used to further gain insight into your difficult child that is key. Our current psychiatrist wants me to have another one done but I just can't afford the cost and insurance won't cover it. She is being careful and thorough before giving a diagnoses. We did luck out with finding her because she did a lot of research regarding the Autism Spectrum during her residency and she suspects that son is Pervasive Developmental Disorder (PDD) and I tend to agree with her.
For me, getting a knowledgeable psychiatrist is key. Unfortunately, I don't have any worthwhile advice on how to get one. So much of it is trial and error. After our former psychiatrist of about a year passed away from Cancer, we were left adrift because all the other recommended psychiatrists were not taking any new patients. And, we had had two psychiatrists before where one required a 90 minute drive and the other mostly specialized in geriatrics, but helped out at a clinic (A nice guy who admitted that Son was more complicated than he had expertise. I appreciated his honesty.). Then, I just happened to call his former psychiatrist's clinic to see if they had gotten a new psychiatrist and they had and we were in!
Anyway, I've blathered on. Medication. For me, if a medication does not provide a noticeable improvement in quality of Son's life, he doesn't take it.
He has taken some where I saw no difference, some where he got meaner (oh boy, and I didn't need that!), and some that helped temporarily but then didn't. But, they did get him through quite a rough patch. He's now on two medications Risperdal (helped him for years), and a new one to help with his ADHD/Impulsiveness, Intuniv which has to be slowly titillated. With the Intuniv I saw the positive impact almost immediately. So, so far so good and as long as it continues without affecting his physical health, he stays on it.
With your boy, I would be very concerned about what he is saying and the suicidal thoughts. Red flag that what he is taking is making things much worse! But weaning off medications is best supervised by a professional. This may not be the case with you, but diagnoses with some kids can be very elusive. It certainly is with my son. He's just never fit neatly into any diagnostic box.
I would definitely begin with Omaha children's hospital and then go from there. Also, a teaching university with a children's hospital is another good source. That is how I found a neuropsychologist when I took son to a pediatric neurologist for a MRI to rule out any brain anomalies.
It can be a long and bumpy road to help our kids, as you most certainly know. Keep pushing and seeking.