Allergies in dogs will usually have symptoms of itching or scratching. It is possible that the food you are feeding him does not agree with him. Science diet makes a Sensitive Stomach formula which you may want to consider gradually switching to and see if it cuts back on his vomiting and occasional diarrhea. If you can't find the Science diet, you can try a lamb and rice formula. Make sure keep a close eye on him, as if he does eat something that he can't pass, he will get an intestinal blockage.
As far as the sneezing goes, dogs can sneeze when they have a cold, they can also sneeze from dust, household cleaners, or anything that can irritate the nasal passages.
I do suggest that you get him checked as well as having his stool cheked to rule out parasites. Best of luck to you and let me know if I can assist you further.