Are you feeding anything other than the vegetables and fruits? Those items should NOT be more than 10% of the diet. You need to offer a hamster food as the main diet.
The vegetables and fruits can easily cause diarrhea.
A vet will probably want to prescribe an antibiotic for the sneezing. You should try adding a tiny bit of gatoraid in the water for fluids and electrolites to help with the dehydration caused by the diarrhea.
Also, start wet tail meds just in case it is wettail.