Why not? Healthcare (drugs, medicines, routine checkups) is a commodity like any other good. It is a scarce service. It is not unlimited. It is absolutely subject to the same economic laws as other goods.
No it's not. If I get in a car wreck and am un/barely conscious, the ambulance is just going to take me to a hospital to get treated. I wouldn't get to call hospitals around and price-check or anything. I wouldn't get to choose what procedures are and aren't performed on me. I'd just be stuck with the bill no matter what.
In fact, it would be pretty hard to do all the research ahead of time and leave detailed instructions behind because hospitals are very secretive and squirrely with their prices.
You could argue that non-essential physician visits are a free market, but emergency medicine definitely isn't.