Is my ethnicity Korean/Mexican or Asian/Hispanic?
. Your cousin is somewhat right and rude. Your cultural ethnicity or nationality is what you have sole citizenship to or multiple citizenship to, but Asian and Hispanic aren’t cultural ethnicities; they are too general and informal as racial ethnicities. Your racial category is the multiracial / biracial category. You is racially mixed with East Asian, European, and Amerindian races. .
Hispania was located in Europe, not mesoamerica
Not if you don't speak the language that goes with it
You can have a DNA test which will indicate your ancestry. Google it for cost.