Hi, thanks for the query to H. C. M. C.
Diarrhea fro one month?  Can't you think, the little has 
dehydration ?
Snot means respiratory infection of course of mild degree, but is there.
Tiny rashes may be due to some antibiotics or also due to mild 
fever ( due to diarrhea).
When there is continue diarrhea , that too from one month long , child will become very irritate , however she is unable to explain to parents. 
Due to diarrhea there is loss of essential minerals from the body of child that is a cause of concern also.
So please first of all , if dehydration take 3-4 bottles of  R L or Elecrolytes according to some doctor.
Till keep watch over the general condition of child and fed her banana , curd, rice of course in small quantity. 
Regards 
Dr. HET