I really don't understand why it being a big country would make things difficult. Any sane person would adjust the needs dependent on state and smaller. Norway's population is about the same as Indiana's, but is more sparsely populated - so some of the same problems as the western states. The benefit of the US being large would be that there are more pools to draw money from. And there are just as many opportunities to save money.
For example, Norway will send a nurse to one's home up to 6 times a day. For free. Even if you live on a cabin on an island or on a mountain. Why? Simply because it is still cheaper than a nursing home, overall. And as a bonus, folks tend to live longer and be happier with life.
As far as immigration goes, it is impossible to compare. But: Not every immigrant gets free healthcare. I get the state healthcare because my immigration status is family immigration - my spouse is Norwegian. If I were here for a job, however, I'd have to pay for health insurance. Of course, they'll still treat someone that pays cash and still treat emergencies. I don't really see how this would be an issue in the US either so long as the laws were set up correctly.
I'll further add that I think the healthcare is a bit different if folks are coming from an EU country - there are agreements in place about health care and things like that. Again, I don't see why this wouldn't be the case in the states.