Here is a screenshot from TPE. Normally this would show the following:
Sunset - the time the sun sets
Civil twilight - defined as when the sun is below the horizon, less than 6 degrees - brightest stars and planets visible
Nautical twilight - defined as when the sun is between 6 and 12 degrees below the horizon - many more stars visible
Astronomical twilight - when the sun is between 12-18 degrees - effectively this is completely dark
And the same things are reversed at sunrise. Twilight is the period after the sun sets, when it is below the horizon, but the sky is not yet dark (or the same period before the sun rises) Civil, nautical and astronomical are increasingly darker (civil = looks dark for driving, nautical = dark enough to navigate from most stars, astronomical = dark enough for most astronomical viewing)
You'll see that nautical and astronomical figures aren't visible, so the sun must not get more than 6 degrees below the horizon.
Nothing magical happens on the 18th. Sun sets about 10:45pm and rises around 4am. But once it sets, it's just below the horizon and the sky will not be completely dark - you'll only see a few of the brightest stars. I would imagine the sky would be no darker than a dark blue.
At the equator, the sun rises and sets very quickly. When it sets, it gets dark pretty fast. This is because it rises very near due east, crosses the entire sky and sets nearly due west, and the day is roughly 12 hours. Nearer the poles, the sun rises more slowly. The sun will rise in the northeast and set in the northwest. So it only covers a much shorter distance in the sky over what is a much longer day than the equator (18 hours!). So it appears to move much more slowly across the sky, and this includes sunrise and sunset.
In most parts of the world, the time between sunset and civil twilight is 15 minutes to a half hour. As you can see, for Rykjavik, it's almost 2 hours. Two hours for the sun to go from 6 degrees below the horizon to the horizon. It will take a similar time to rise 6 degrees above the horizon. So you should get long, slow sunrises and sunsets and plenty of blue and golden hours.
Compare that to the equator, where the sun covers 180 degress in 12 hours, or 15 degrees per hour. In Iceland in May, more like 6 degrees in 2 hours. Quite a difference.
After May 17th, the sun will continue to set later and rise earlier, until the summer solstice around June 21st.