True, but the point is, usually multiple numbers will route you to emergency services.
For example, in the UK 999 and 112 are both officially recognized numbers. And GSM mobile phones support most of the widespread codes (911, 112, 119, 999…).
But…I’m having trouble finding a cite that 911 works in most countries (landlines).
Either my google-fu is not up to it, or I was simply wrong.
If it’s the latter, it’s about time 
Depends on your problem. In the OP’s hypothetical, it’s probably the same thing. If you break a bone, as long as it’s not a compound fracture they’ll put you in a splint and/or give you pain meds, and tell you to see an orthopedics specialist.
No, it does not depend on the problem. I think when most people say hospitals have an obligation to treat you they don’t mean the hospital cannot transfer or discharge you until you are 100%. I don’t think many believe that these days.
When you are stabilizing a patient, you are treating a patient.
mmm