is this word common in the USA? I sometimes hear "in another country".
The word "abroad" is fairly common. A fluent English speaker would certainly know it.
Checking a couple of word frequency lists, it appears the word "country" is around #200, and the word "abroad" around #2000.
Are my examples correct?
Not exactly, no.
The most usual ways to say "abroad" are "traveled abroad", "studied abroad", "went abroad". Here are examples from the deepl.com translation page:
en el extranjero (adv.)
- I want to study abroad. Quiero estudiar en el extranjero.
- My brother returned home after two years abroad. Mi hermano volvió a casa después de dos años en el extranjero.
- I must grasp the opportunity to study abroad. Debo aprovechar la oportunidad de estudiar en el extranjero.
- I want to travel abroad and learn about other cultures. Quiero viajar fuera y aprender sobre otras culturas.