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).