You have a lot to take in but you are at the right place for help. Gerd can be and is a very tricky illness to deal with as the symptoms and issues can change from time to time, it's a little different for everyone. I will try to touch base on several items for you.
Medical care......I'm sure your ped doc is good, however I can't say it enough, if it were my child I would ask for a referal to a childrens hospital. (I had to due this with 2 of my own children for medical issues) Gerd can be very hard to treat, so it's important to receive the best care and advise you can get. Children hospitals specialize in hard to treat medical issues for children, it's just a matter of using the right tool for the right job.
Anti-reflux meds.....there's ppi's (nexium, etc.) h2 blockers (zantec, etc.) they both lower the acid level to prevent damage. However, like rjdriver said these make it harder for the body to break down the goodness your body needs. IMO, you have to be careful with supplements and children as thier bodies can't handle what adults can. For now a childrens multi-vitamin and maybe a mag/cal with vit D supplement would be a good start. You don't want to over tax your little guy. Carafate is a med that coats and protects the throat and tummy but can be hard to take because of timing issues around food, supplements and meds. Tums can be used also, I believe one of the brands have came out with fruity tasty chewables.
Sleeping.....everyone is different but normally on your back or left side is best but sleeping with your body (waist up not just his head) on an incline is a must. Most due anywhere from 5 to 8 inches. You can buy beds that incline or prop up the head of the bed with books if money is tight. Just putting pillows under his head is better than nothing but will not work as you need it to in the long run.
Surgery......IMO, this should be viewed as a last resort after a long period of trying other things and they didn't work. "If" you ever do, the nissen is the gold standard for gerd, linx is just to new and everything else doesn't have good enough results. Again if you ever find yourself going down that road, NEVER let any doctor near your child for surgery unless they are from a childrens hospital!
Diet.....This one is hard for adults, so that makes it even more of a pain to deal with in children. For starters, shame on your ped for the lack of advise (goes back to my childrens hospital comment), about avoiding fruits and soda. It's a lot more detailed than that and it really changes from person to person but there are some of the main bad guys. Anything acidic in nature is a no no, a lot of fruits may or may not cause issues. Oranges almost always does but melons (watermelon, canople, etc.) are normally very soothing and easy on the tummy. You want to avoid foods that are hard to digest, as they will just sit in your tummy and cause more issues. Green veggies are good for you but some ppl have problems with them and others don't. Now for the bad news of food.....you really need to avoid all soda, choc., greasey/spicy, fast food, red sauces (pizza etc), red meat (hard to digest), chips, candy (laffy taffy etc). Yes, it's hard but in time you'll find relacements for most of these things. Some examples....replace soda with fruit punch, hamburger with chicken, turkey, or fish, it just takes time to see what causes issues and what doesn't.
Supplements......there's tons of "snake oil" ppl will push, so lets focus on the proven healers and helpers. Again with children I always like to be careful. Manka honey is the best know natural healer there is, like rjdriver said it can get pricey. With manuka the higher the active + the better it is, I use wedderspoon active 16 + brand. I mix it with warm water and make a tasty soothing honey tea. Aloe vera can help heal and sooth also, there are favored brands and you really only need a couple oz a couple times a day. A childrens multi-vit chewable and maybe a cal/mag with vit D, that is probally good enough to start with for now.
Barretts......did they mention if it was long or short segments? What was the length and location? Was there any dysplasia? Does he have a hiatil hernia?(This goes back to the right doc for the right job) Some doc confuse were the throat ends and the tummy starts thus they biopsy the tummy that has hernia instead of the throat giving a false positive for barretts. This happens a lot more than ppl think.
Well you have a lot to take in but this will help you understand what you are dealing with. We are all here to help each other so don't be afraid to ask.
Good luck and stay strong!