Iceland is so far north that depending where you are in the country, you get significantly different results. Compare the following chart from gaisma.com for Reykjavík, which is in the south:

With this chart, for Ísafjörður in the north:

As you can see, between mid June and the beginning of July, the sun never actually sets in the north of Iceland, however in Reykjavík it DOES set, but it stays light enough to remain twilight throughout sundown (the pink area). At this latitude, there would be little to no 'blue hour' as it rarely gets dark enough, even in Reykjavík.

To answer your final question, the best time to go for as much golden hour as possible would be between mid May and late July, as this is when you get the most twilight (as much as 5 hours), and as it occurs in the early hours of the morning, you will not be competing with tourists for the best scenes.

Note that as Iceland is not on the meridian, but uses GMT throughout the year, the sun's lowest point occurs at around 1:30 am, not midnight, so you can still get 'midnight sun' in Reykjavík around the summer solstice, even though the sun does eventually set (around 12:05 am).

Oh, and have a great time. Iceland is a wonderful country!