"The river is flowing down the side of the mountain."
Translation:Floden rinner nedför bergets sida.
The only difference is that "nedför" is more formal. You also have "ned" and ner", where "ner" is much more common. In compound words though, "ned" is often preferred to "ner" and sometimes even the only option, e.g. nedlåtande (patronizing, condescending).
None really. The former is written language, and the latter is spoken language. Both can be used in writing however.
Why is "sidan" considered a mistake here, despite the english being "the side" ?
You mean why it is wrong to write "Floden rinner nedför bergets sidan."? That is not possible, as "bergets" is the genitive form and it is not possible to use a definitive form after a genitive form. That would be like saying "The river is flowing down the mountain's THE side." in English. Which is not possible, too, as far as I know. (I'm from Germany. English is therefore only my second language.)
From what I have learnt here, the word “ned” cannot take an object directly, while you can add an object immediately after the word “nedför”.
However, “ned” can take an object if you add a preposition after the word “ned”. For example:
Katten hoppade ned från bordet.
The cat jumped down from the table.
Mannen går ned på berget.
The man is walking down the mountain.
You can use the word “nedför” for the example above without changing the meaning:
Mannen går nedför berget.