My dog has diarrhea and a rash near her nose. Could they be related?

I doubt they were caused by the same thing but that doesn't mean stress or being sick didn't affect another part of the body to make it more susceptible to a disease. There is nothing to put on her mouth since she will just lick it off. You can try boiled chicken and rice and add canine probiotics to the food to help with the diarrhea until you can get her seen. She will likely need a fecal exam and a skin sample by a vet since it has been going on for so long