turtlenecks are cute and sexy but it depends on the type and also how the guy looks lolz
anyway, back to the original question: no, it's not cold all the time in Korea. You just choose the dramas that were filmed or had the plot in winter/autumn.
Korea has 4 seasons (just like my country lolz): spring, summer, autumn and winter. And as a matter of fact, now it's summer and there are 33 Celsius degrees over there. There are 32 Celsius degrees today here in my country. :D