I have some info being that I'm going through the same thing. I start out with rotten egg smelling "sulphur burps", then within a few hours start throwing up which also smells the same and my stomach starts to feel like I'm getting stabbed right above my belly button!
The good news is I have a list of possible things it could be.. the bad news is you'll have to get some blood work or other tests to find out which you're suffering from.
What they think I have is Celiac Disease, which is pretty much an allergy to gluten. They did a blood test on me today to see if that's it, and I'll get the results next week. If it's Celiac, a gluten free diet will help or cure it because it's the Gluten that's imflamming or breaking down the tissue in your stomach/ intestines which causes the smell.
Another disease this is a symptom of is Crohns Disease, which I don't have much info on, but I believe it's similar to Celiac's and that there are tests for that also. I also believe that a special diet is the cure/ help for that as well.
Another possibility is that you have a bacteria.. either H. Pylori OR Giardia.. both are tested by getting an endoscopy (where they put you under mild anesthesia and run a small camera down your throat into your stomach). They take a biopsy and test for those. These bacterias can be cured with special antibiotics.
The last possibility that I know of is something called "gastropariesis", which is where your bowels move very slowly and the food you eat doesn't digest as fast as normal. To test for that, they have you eat something like scrambled eggs with a small amount of radioactive stuff in it, then they put you under a machine that follows where the radioactive material is in your system and how fast it's going through your body. It takes a few hours. The thing about this disease is that your stool doesn't pass through your body fast enough, so it's kind of like it backs up, which is why your burps and vomit smell like rotten eggs (or farts). Pretty gross, but I'm not judging since it's possible that's what I might have! lol They can cure this with medication like Reglan to make your bowels move faster.
I hope all this info will help some people out there and you'll be able to talk to your doc about the possible diseases/ problems to test for so that you can get it cured asap!!!! I just started my testing and I'm hoping for the same! Good luck!!