“America” refers to two continents, containing a wide assortment of nations, peoples, and languages. If you’re referring to The United States of America, abbreviated as “the US.” However, it is perfectly acceptable to call US residents “Americans.”
Don’t use “The States.” It’s far too casual, and it’s inaccurate: several parts of the US (eg, Guam, Puerto Rico, Washington, DC) do not have official statehood.
Capitalize “the” in “The United States of America,” but not in “the US.”
“The US” is a noun. “US” is an adjective.
It’s counterintuitive, but “The United States of America,” and all of its short versions, is a singular noun. Ask a historian why.