When I read this word for the first time my immediate association was "Umgebung" (I am a native German). So, yes, I would use this word in the same way as "Umgebung".
Kind of, but this means area, place or district, whereas omgivelser means surroundings, i.e. what's surrounding a particular place.
"The areas" isn't a phrase i would use in isolation in English. Do you use it like "environs"?