You don't HAVE to have an article here in english when it is not part of a sentence.
Why do I need 'a' here?
Well... if it was just one noun phrase (dziewczynka) we would definitely accept it without an article. here... I'm not so sure, but ok, it's not a sentence, after all. Will be allowed now.
I think because it's singular
But why do I need an article at all?
Because that's how the English language is?
If they're being used in a sentence, generally yes. But just standing by themselves (as they are here), not necessarily.
Shouldn't Dziewczyna be valid here too? Both mean girl, one just isn't diminutive.
Of course, "dziewczyna" should have worked, it's accepted.
Is there any techniques I can use to remember the word 'girl'?
Are* sorry for inccorect grammar.
'The girl and a boy' was my only correct option... So the noun in its own can be definite or indefinite?
The main answer is "a girl and a boy", but all variants seem acceptable. As Polish doesn't use articles, in most sentences (and this is just a phrase, so even less context) both versions would be correct.