Why cant it be "the man sees the girl" how you know if its the or a?
It can.
You do not know out of context if it is "the" or "a" in high valyrian.
what are the differences between riñe, rina and riñi?
Riña is the nominative case (who? - a/the girl). Riñe is the accusative case (to see whom? - a/the girl). Riñi is the plural form (who? - (the) girls).