Just to back up with Daniel said, yes, we say we have a "runny nose"; I've never heard of a "running stomach"; it's diarrhea but much more commonly called "the runs" (because you're running to the bathroom! lol). It's somewhat of a slang term, but very much understood by all; I really wouldn't even consider it slang, really.

1. In my experience, people tend to say "runny nose"* rather than "running nose", although someone's nose "is running". 2. Diarrhia is called "the runs" as a noun, but there is no corresponding verb form based on "to run". *I've never seen this in print; so, it's probably a nonstandard variantion.

It's a "runny nose", not running! And --- she has the runs = she has diarrhea