So, if I understand the concept of "k" correctly, this would have the sense of "this town is near in the direction of Moscow"? But isn't every town near Moscow possibly in its direction? In other words, what is the difference between "этот город близко Москвы" and "этот город близко к Москве"?
You cannot use «близко Москвы», it is as simple as that. «Близко от» and «близко к» will work (they use the Genitive and the Dative respectively, just as от and к normally would).
Think of «близко» as "close". You can be close but you cannot be "close Moscow"—English uses "close to" in this case.
Ok. I have the same question as Jenny: what is the difference between близко+к+Dat. and близко+от+Gen. and when should you use one over the other? Could you say «Этот голод близко от Москвы»? In that case, what difference meaning would it convey compared to «Этот город близко к Москве»?
That's wrong. Near and near to mean exactly the same.
"Возле" is usually closer than "близко", but it's not set in stone. Just like English has several prepositions denoting close proximity to something ("by", "near", "close to", "next to", "alongside" and probably more) without explicitly defined difference (even though they are different), so does Russian ("близко", "возле", "около", "рядом", etc). "Этот город возле Москвы" is possible but it can only mean a town that is right next to it.
You have to remember, though, that different prepositions demand different cases, so sometimes the system rejects the answer not because the preposition is wrong, but because the case of the noun doesn't suit it.
It doesn't (and even in English the formal criteria are not set in stone). In English translation the word "city" usually applies to urban settlements of sufficiently large population, like over 100–150 thousand. Which makes one think how a town is different from a city: in some countries a settlement with way fewer people might be classified as a city (at least, in English).
On the other hand, Russian has two words for a village, деревня and село—which are not clearly defined either (село used to mean, technically, a larger settlement).
Thanks. Out of curiosity, what is the origin of the non-standard plurals in Russian? Are they surviving archaisms from the grammar of earlier eras (as is the case with many English non-standard plural forms)? Or do they arise from when foreign words have been imported into Russian?
I think, in Old Russian those words belonged to another declension. Then the number of declensions has decreased and they became irregular.
Another version is that "город" has something to do with the Old Church Slavonic "градъ" which was "града" in dual, according to Wiktionary (but I don't know if the final "а" was stressed).
Yeah, that structure is also a possibility but in a different context. You can encounter it in "близок к Богу".
These days we rarely use this adjective's short form (близок/близка/близко/близки) spatially but we often use it in the abstract sense, i.e. "almost, quite similar to" («показатель близок к 100 %»).