No. Not related to teething.
Likely not related. Teething will cause some some children to become more irritable and uncomfortable. You will likely also see an increase in saliva flow and the desire to chew on things (like their toys). It should not cause any rash however. The two are likely unrelated. I can't comment on the rash as it is out of my scope, but i can say with confidence they are not likely related. Best of luck to the little one!