It can mean that of an object but when talking of a person it means the same as turn around. Don't be so petty. It is spoiling the enjoyment of this course. The vast majority of people doing this course are doing it to improve their Swedish not to get bogged down in pedantic nuances of English
Actually, no. Both Google and Wiktionary give one definition of "round" as "so as to rotate and face in the opposite direction", e.g. "he swung round to face her". In British English at least, one can "look round" or "turn round" without having anything to do with circular objects.
Not really. You can say something like "Let's take a turn around the garden." That's a rather old-fashioned way of saying you're going for a short walk, and you probably would be looking at the thing you're walking round.
If you said you wanted to "turn around the villa" I might think you were rearranging all the furniture.
I don't know if there are any similar expressions in Swedish.
Runt means "round" as in "a circle is round", it means "around" as in "he turns around".
Omkring means "about" in the sense of "He has about 5 minutes left (though you might use "ungefär" there). It also means "about" in the sense of "wandering about the castle grounds" (slightly archaic in English).
You combine the two in "runt omkring" meaning "all around" or "in the area".
"Området runt omkring arenan har aldeles för få barer." (The area around the arena has far too few bars.)
"Jag tittar alltid runt omkring innan jag knappar in mitt PIN för att se till att ingen tittar." (I always look around before I enter my PIN to make sure no one is watching.)