Thank you for your query,
The purple marks on cheeks due to dry skin or due some allergic rash
Nothing to worry and you can apply one emollient
like dermadew or cetaphil over cheek skin and this rash is not related to the eruption of the teeth.
On and off diarrhea is due to some contamination during that time and to be treated with ORS, Pedialyte, probiotic
and zinc after consulting your Pediatrician
Hope I have answered your query.