Belching that tastes and smells like sulphur (rotton eggs) usually followed by a painful stomach ache. This lasts till I puke up. What can this be?

I am not sure. At all mainly since you are so young. Your complaint is usually the case of motility namely gastric emptying problems. Other option is infestation with a single cell parasite giardia lamblia but this goes with diarrhea you do not mention. I suggest see a GI specialist to check your stomach emptying. If that would be confirmed, blood test r/o an auto immune condition is in place.
Giardiasis. The "purple burps" are famous. There are other possibilities.