Short answers, pulled from the story.
Ingólfur Arnarson arrived on the southern shore of Faxaflói bay in 874. He cast his high seat pillars overboard and settled where they were found after three years, giving the place its name meaning smoke and bay from steam rising from hot springs.
Skúli Magnússon led the movement to establish a trading town with houses built in the 1750s housing the wool industry. The re-establishment of Alþingi in Reykjavík in 1845 effectively made the city Iceland's capital.
Four British warships approached Reykjavík on the morning of the 10th of May 1940 following German occupation of Denmark and Norway. Allied forces completed their occupation within hours without armed resistance while foreign soldiers numbered about the same as the local population.
Reykjavík sits at latitude 64°08′ N making it the world's northernmost capital of a sovereign state. The North Atlantic Current moderates climate creating mild winters and cool summers where temperatures rarely drop below freezing.
Volcanic activity provides all Reykjavík and surrounding areas with geothermal district heating systems for residential and industrial districts. Natural hot water heats roughly 90% of all buildings across Iceland from combined heat and power plants plus lower temperature geothermal fields including Nesjavellir and Hellisheiði CHP plants.