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Yes. Doctor's from any country can work in the U.S. as long as they are in the country legally and have been approved by the appropriate medical associations to work there.
they are called overseas worker.
If you are flying within the United States, you do not need a passport. If you are flying into the United States from another country, or out of the United States into another country, you need to present a passport at customs.
How do you sponsor a person coming from another country to the US
This varies from one staffer to another, so there is no way of telling how long unless you ask them.
It makes no difference how long a US citizen of ANY age is out of the country. "Once a US Citizen, always a US Citizen." -UNLESS- you renounce your US citizenship and become a citizen of another country. However, since the question is about a 'minor' child that eventuality should not arise.
report to the US consulate or embassy in that country.
Typically, a Working Visa. The process varies by country. Usually, you'll have to show that you're bringing in skills that are scarce in the country.
The US does not own any countries, The US itself is a country. A country can't own another country.
Magic Jack can be used in any country as long as you have an internet connection.
No
Technically, you can return to your country after overstaying your visa then return to the US. However, you must not that if you have been illegally present in the US for a long time, you will be barred from returning to the US for 3 or 10 years.