Is "the USA" singular or plural?

Short answer: in contemporary English, both USA and the long form United States of America are treated as singular nouns.

Long answer: Language Log has documented this in great detail. In the 18th and much of the 19th centuries United States was treated as plural, but in the latter half of the 19th century the singular usage became more common. Today, the singular usage is the only accepted usage, except for the case of a few fixed phrases. In fact, "in 1902 article in the Washington Post reported that Foster's work (which evidently was reprinted as a pamphlet) had persuaded the House of Representative's Committee on Revision of the Laws to rule that the United States should be treated as singular, not plural."

The United States of America own this domain

To me this sounds a little bit awkward, as the United States of America is one entity. Actually, it's likely because the pluralism is buried in the middle of the term.

If you were to use simply The United States... I would accept either own or owns, depending on what you're trying to emphasize: the collection of states as one entity or the collection as a group of states.

Both "USA" and "The United States of America" are a single proper noun. They are names. I don't believe you can point to a word within a name to call the name a plural. Both names refer to a single entity. They should be followed by the singular form.

The exception is in some British English where singular nouns representing collectives (companies, teams, governments, etc.) are treated as plural. Some reading on that: http://languagelog.ldc.upenn.edu/nll/?p=877

I think you chose whichever sounds best to you, as there's clearly not right or wrong. Style guides, for example those internally used by the BBC, suggest you pick one and stick with it, at least for that article/feature.