Short answers, pulled from the story.
The Christianization of Scandinavia took place between the 9th and 12th centuries. The earliest recorded signs were Ansgar's construction of churches in Birka and Hedeby in the 830s, and the conversion of Scandinavian kings occurred over the period 960-1020.
Harald Klak, King of Jutland, was the first Danish king to convert to Christianity. He was baptized in Ingelheim am Rhein in 826 along with his family and four hundred Danes, in exchange for Emperor Louis I's political support.
Harald Bluetooth converted to Christianity around 960, reportedly after the Frisian monk Poppo held a fire-heated lump of iron without injury. He also converted for political reasons: to preserve his independence from the Germans and to gain the practical benefits of literate missionary advisors and access to Christian wealth.
Iceland converted to Christianity by a decision of the Althing in 1000 AD. The pagan leader Þorgeir Ljósvetningagoði was given the role of arbitrator; after a day and a night of meditation, he ruled that Iceland would convert as a whole while private pagan worship would be tolerated.
Ansgar was the central early missionary to Scandinavia. Assigned to accompany the exiled Harald Klak back to Denmark in 826, he later traveled to Birka in 829 and established a small Christian community. He became Archbishop of Hamburg when the archdiocese was founded in 831, and returned to Birka in 854 and Denmark in 860 to rebuild Christian communities after pagan setbacks.
The spread of Christianity among ordinary Scandinavians took much longer than the formal conversion of kings. Archaeological excavations at burial sites on the island of Lovön near Stockholm show the actual Christianization of the population took at least 150 to 200 years. Some Sámi people remained unconverted until the 18th century.