Let me add to the other comments that "America" is a noun referring to the country, while American is an adjective, or a noun referring to the people. I am an American; I'm from America. More technically, I'm from the United States of America in the continent of North America.

If asked "where do you live," the answer that pops most quickly to my mind is "I live in the United States." I might say "I live in the US." There's really no difference in meaning between "the US" and "the USA." The full, official name of my country is "the United States of America." So, "United States" and "US" are slightly less formal. "USA" stands for the more formal full name. If someone says "USA" I wonder for a split-second why they are bothering to add the "A." It carries a slight feeling of patriotism. "American" is an adjective. The way you used it is incorrect. You can say "the American flag," "American cars," "American history," and so on, the way you could say "the Chinese flag," "British cars," or "Guatemalan coffee." We almost need to use the adjective "American," because in English there isn't any good adjective for "United States." You can't say "United Statesian" or "US-ian." "American" is the only adjectival form you will find in dictionaries. We often use "America" to refer to our country when we are being reverent or patriotic. Avoid it if you are just stating an unemotional fact. "America" and "American" are troublesome. To us, "America" is just a way to shorten "United States of America." However, "the Americas" means "North America and South America," and "America" can mean the same thing. Other people in the Americas--I think Chile and Argentina in particular?--regard themselves and everyone in the Americas as "American" and dislike it when people call the United States "America." (In Internet forums with an international membership, I've actually seen people use the abbreviation "US-ian," because they know that using "American" annoys some readers).

Yes, all 3 answers are correct. The US is just a shortened USA, it means the same. American is almost always associated with USA as well, however technically, American can mean anywhere on the continent, depending on the context. However in reality nobody calls themselves Americans except people in the USA.

“I live in the US” is more natural and neutral than the other options. There are also other possibilities such as “The States” and “US of A”. But you can’t go wrong with “US” or “United States”.

All the same meaning. You can say I live in the US or in the USA. America generally refers to the US or all of the Americas combined.