When I sneeze I get a pain in my lower stomach. Infection?

No...muscle. When you sneeze, you contract a number of muscles in your chest and your abdomen or stomach area. The contraction of these muscles is usually quite vigorous. The pain you are having is most likely from rapid and vigorous muscle contraction during a normal sneeze.

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Is h pylori stomach infection contagious? Can it be passed on by sliva, sneezing etc

Yes, it is. presumably contagious. Mode of transmission is usually via contaminated water or foods. It can presumably transmitted by saliva, but this is more theoretical. Read more...