Why is my kitten vomiting and having diarrhea after being neutered?

It is possible that he could have had a reaction to the anesthesia or just the stress of being neutered. Or it could be completely coincidental and it could be secondary to something else like parasites, foreign body, a sudden change in food, pancreatitis, an underlying infection, and much more. I would try to feed him some plain boiled boneless and skinless chicken and white rice. I would also try to start him on some probiotics. Your vet should be able to provide these or give you some recommendations. You can try some plain sugar free yogurt but the probiotics in this isn't as good as the cat probiotics because we have different bacteria in our gut than in cats. The next step should be to have him re-examined and likely have some bloodwork done. I would also make sure the vet has ruled out an parasites including giardia which is difficult to find on a regular fecal test. If your vet doesn't think there is a foreign body, they will be able to provide him with a good anti-nausea medicine to help with the vomiting. I hope this helps!