As for national borders being unethical by nature, if that logically follows then why not?
I certainly don't consider ethics a source of "gotcha" questions, but of real conundrums.
As regards providing free healthcare to anyone regardless of country, that seems next to impossible without providing the infrastructure necessary within the country they live, which would indeed be very difficult to work out. It seems like a good goal to work towards - everyone regardless of nation having access to reasonable quality healthcare.
Not really. A huge amount of charity money supposedly going to the poor in other countries is actually used to bribe officials in those countries to allow the aid to be delivered. Any attempt to give healthcare to people in those countries would also be seen as foreign interference, and suffer the same problems.