Would there really only be one temple per city or is that just the result of the world presented in-game being scaled down do to hardware limitations? Are temples the equivalent of hospitals in Tamriel or would there be a separate institute for that? Where would someone go if they broke their arm or something? Are there things like doctors?

It's probably a safe bet that Restoration and Alchemy are the primary forms of medical treatment. In "Real Tamriel", obviously people don't actually have an HP bar to indicate how close to death they are, so the various Restoration spells and effects should be assumed to be actually repairing physical bodily damage. Basic spells that heal minor cuts, burns, and bruises, stronger ones to mend broken bones and deeper gashes, and replace spilled blood, and the strongest that will re-grow damaged organs, reconnect severed limbs, etc.