"Makes poems" sounds strange to me, although English it's not my primary language. Should "writes poems" be accepted?
You are right that it sounds strange in English but the Latin word 'facit' translates as he/she makes. 'Writes' would be 'scribit'.
Would "poetry" be a valid translation for "carmina"?
Not really. It means poems or songs or spells.
Is "carmina" In accusative case?