"She lives in Argentina."
Translation:Mae hi'n byw yn yr Ariannin.
"Yr Ariannin" is just the Welsh word for "Argentina" with no need to translate the definite article before it, this is used for other countries too including "Yr Eifft" (Egypt) "Yr Alban" (Scotland) and "Yr Eidal" (Italy), none of which use the definite Article in English.
Could you say 'byw'n' like in the case of 'hi'n'?
No -- this is a different yn.
The preposition yn meaning "in" doesn't contract to 'n, unlike the particle yn that links a form of bod to a predicate (mae hi'n hapus; mae hi'n athrawes dda) or the particle yn that turns an adjective into an adverb (os gwelwch chi'n dda).