Translation:A hundred new buildings are being built in the town in June.
I'd say by on its own is much closer to what you call a "town" in English than to a city. It's pretty much a catch-all term, though. If you want to talk about a city (being an especially large settlement), you can use the term storby. Conversely there's småby for smaller towns.
Småby is not a much used word on its own, but can be seen in its plural form, småbyer. You would more often see the word Landsby which probably would fit Village. The term Købstad is outdated but still used. If I understand it correctly it is to be compared to a borough, a town with special privileges given to it (by the King).