Would one say this referring to using a stethoscope to listen to one's chest, or is this more like a woman laying with her head on your chest?
The English sentence is off. Either you use a different verb or a different preposition. If you stress the act, you‘d say put; if you stressed the resulting position, you‘d use on or maybe touching.
What's the etymology of brystkasse
Not sure, but we say Brustkorb in German too, literally breast basket. I imagine it simply describes the basket-like bone structure.