Street plowing is only really happening in the big Swedish cities and on highways anyway. On the countryside, everyone has studded tires on their cars, so they only put gravel on the roads.
Edit: As several answers point out, snow is being cleared, just not all the way to the asphalt. Which is what I meant as well, apologies if that wasn't clear.
Of course country roads are plowed too if need be. Every small city where there are roads that needs to be plowed are plowing their roads. Depending on where in the country you are and how common it is with snow, roads may be plowed sooner or later. It's definitely not only the big cities.
No, it's all cleared. You can't drive on the roads if the ice is still there in spring. You will sink 2 dm down and when that freezes over again, the road is only usable by tanks. All roads that needs it will be scraped of ice sooner or later.
I have been living in north Sweden all my life, in tiny villages with 100 people and cities with 100.000 people. They all do it. It might even be better done in small villages because they will use local companies that live there and does their hometown on contract every year.
Roads are definitely cleared from snow outside of cities. This is an example from the middle of nowhere, where signs of clearing can be seen: https://goo.gl/maps/1b7Aj8pb3ejsQPHs7
However, in the northern parts/and or rural areas, you have a layer of packed snow/ice during the winter season. Snow is still cleared regularly that fall on that layer, but the underlying asphalt/gravel isn’t revealed during winter.
I remember driving from Kiruna to Abisko National Park (read: from a tiny town in the far north of Sweden to a national park in the middle of nowhere) and there were huge tractors cleaning the snow in both directions almost as soon as the snow had passed a few inches.
From the point of view of an American, for example, that's like a road in the north of Alaska, connecting the only two more or less significant towns in something like 100 miles... the area around Kiruna probably has less than one million people in a 1,000 mile radius, it's really a low population area we're talking about.
True, we are only 9 million people total and the north is mostly uninhabited. I totally see how that feels tiny. Anyway by your description I pictured a small winding 1-line 40mph road, not a 55mph banked 2lane road
Edit: As several answers point out, snow is being cleared, just not all the way to the asphalt. Which is what I meant as well, apologies if that wasn't clear.