I know it's so hard with an infant when you are learning your way around parenting. It's not easy and there are many worries.
You don't go to the ER for a rash. That will cost a fortune and will have you sitting and waiting while really sick kids get seen first. You'll sit with a bunch of sick kids and wind up with bigger problems! You see the pediatrician, or you see a dermatologist. The ER is for a major injury or a very high fever, and only when directed by the pediatrician - always start with the doctor first.
Babies often get "baby acne" - is that what it is? It goes away after a month or so - normal. Baby drool can be a problem, so can baby washes and lotions. I'd be surprised at Cortisone 10 for a kid who could put his hands in it and then put his fingers in his mouth.
And how does a 2 month old rub his cheeks on furniture? He's not that mobile. If you put him down on the floor on a blanket, on his back, none of this will happen. If you put him on his stomach on the coach and he tries to lift his head, yes, his cheeks will rub. So just don't do that. And a 2 month old should not be anywhere near a pillow - yes, suffocation is a risk. So, no pillows!
Is he breast fed or formula fed? That can make a difference. But if they are 2 small patches, I'd think you could let it go for now or look into the other causes mentioned.
I'd suggest that you get a good baby medical book - as the pediatrician for recommendations or ask the librarian at your public library.