They are called 'International Borders'
borders
The rules made for making relations with foreign countries is called foreign policy
parts of countries surrounded by foreign territory is enclave or enclaves.
Countries do not have affairs in the way you imply in your question. Countries have foreign relations with other countries and the department that deals with these relationships is called the foreign affairs department.A foreign affairs department will have relations with all countries. some of which will be ambassadorial relationships.
In general terms a professional who does this is called a "diplomat".
China is one of the countries that borders India. Also, another country that borders India is called Pakistan. However, Pakistan and China do not necessarily get along.
They are called Embassies or an Embassy:)
foreign interference
President Wilson's plan called for borders of countries to be determined by ethnic groups living in the area.
US cellphones will get signals in foreign countries if and only if your provider has collaboration with any provider in that company in that foreign country.This is called "International Roaming".
It is called imperialism
outsourcing