Don't know about Farsi, but in Russian you would have to declare your knowledge and specificity of the subject. You can transmit some of that information by changing word emphasis, but it's not very clear.

As for your sentences I would translate them as:

1. The girl downstairs was playing violin → Девушка [girl] внизу [downstairs] играла [was playing] на скрипке [violin].
2. A girl downstairs was playing violin - Внизу [downstairs] девушка [girl] играла [was playing] на скрипке [violin].

The first one puts more emphasis on the "girl". The second one puts more emphasis on "downstairs", almost suggesting that the identity of the girl doesn't matter, kinda "some girl who happened to be downstairs". But it is not as explicit as a definite/indefinite article for sure.

If you want to make it certain that the identity of the girl doesn't matter, you could use an adjective to that extent: Внизу [downstairs] какая-то девушка [some girl] играла [was playing] на скрипке [violin].

If you want to make it certain that both you and the person you're talking to know about the girl, you can use a determiner: Та девушка [that girl] внизу [downstairs] играла [was playing] на скрипке [violin].

We do the opposite. If something is known we just say it.

"The girl(1) is playing(2) violin(3) downstairs(4)"

دختر(۱) در طبقه‌ی پایین(۴) ویالن(۳) مینوازد(۲)

To refer to an unknown thing we add a prefix ی to the thing

A girl -> دختری
The girl -> دختر

I guess the is implied, count is explicit. A thing becomes one thing and the thing becomes thing.