"The Bahamas" is an exception, perhaps because it is plural, much like "the United States". Most countries do not start with "The". For instance, Germany, France, Iran. They happen to be singular. However, I think that most people leave off "The" when referencing United Arab Emirates. Many people say "China" but also "The People's Republic of China."
As to "the" before a person, people's names just do not use "the" in singular. However, "The Obamas" in plural refers to the entire family, i.e. the four Obamas living in the White House.
For the question of articles used with county names, there appears to be individual convention, but not a rule. Someone who studies English rather than simply living it may come up with a better answer.