Can Russian preposition "на" be used with any other country name besides Ukraine ("Украина")? Or is it like only in case with Ukraine it is correct to say "на" instead of "в" (that is "на Украине" instead of "в Украине")?
It's a legal preposition for locations, but rarely used for countries, rather for some toponyms, e.g. на Волге, because it's also на реке - at river. Cuba and Taiwan are islands and "on" island is to be на острове. Ukraine only seems to be a rare case, but it isn't.. due the old meaning of country's name. It was derived from term оукраина, that existed in medieval times. Basically it is something close to a fief. In Imperial time name of Ukraine was
Малороссия, but it was generally called
украина without specified name (a common term at that time, e.g. Ростовская украина), which slowly took over as a name of province and was capitalized. By the way, the theory that name means окраина (province, border, distant area) is considered methodologically incorrect. Regardless, both locations would require preposition на.