I will just start out that being a first time mom is the most stressful thing in the world that not one person can prepare you for. Even if someone tried you wouldn't get it until that baby was here.
There are a couple of things I would try:
First, I would go to another doc. I am not sure you need a specialist but you could certainly use a second opinion. You also want to bring him when he is hungry so you can show the doctor what is going on. Sometimes the doctors do not always understand that severity of the situation.
Second, keep in mind that your baby is flourishing, which means he is getting enough food. I have three kids and one of them acted a bit like this. although it is stressful, it may be that he is just trying to get a handle on things and it will take a while for him to calm down about eating.
Third, one of the toughest things we do to our babies is switch formula a ton of times. I know how hard it is not to when your baby is so upset but you do want to give one type of formula a solid 2 weeks before you make a judgement of changing it. I also know that although soy is a solution for reflux, babies typically hate it and fuss about eating it. My one child that had this problem never would take soy, I kept him on regular formula and had a prescription I gave him just before he ate every time. I wish I could remember the name of the script - sorry - but it sure helped.
Unfortunately, babies are SO hard to diagnose since they cannot tell us what is wrong. Doctors do their best to give direction but ultimately as parents we have to wait a lot of these things out. My son wasn't truly better where I wasn't worried and stressed until he was about 8 or 9 months old. I felt the same thing as you, massive frustration, anger at the docs, and finally depression.
Your baby will be fine, my guess is he is just taking a while to calm down and indeed the cord around his neck could have played a part. Just remember that your emotions and frustration transfer directly to your little guy, he can feel what you feel. So, the more upset you are the more upset he will be.
Good luck!