Is there a difference in Irish between 'on' and 'at'? (So would 'The farmer works on the farm' be interchangeable with 'the farmer works at the farm'? Or 'the girls are on the beach' with 'the girls are at the beach'? I tend to use them interchangeably in English but I've been marked wrong twice with that sort of change.)
There is a difference between "on" and "at", just as there is in English. The fact that you use them interchangeably in some circumstances doesn't mean that the two words are equivalent. While prepositions don't often translate directly, always go for the direct translation unless you are quite sure that it is wrong.
The grammatical gender of a word has nothing to do with the person that a word is applied to. All "job" words that end in eoir or óir are masculine, and the teacher, the lawyer and the farmer are just an múinteoir, an dlíodóir and an feirmeoir, whether they are a man or a woman, just as "the girl" is an cailín.