Short answers, pulled from the story.
Gardar Svavarsson was the first Viking to sight Iceland after going off course due to harsh conditions while sailing from Norway to the Faroe Islands. His reports led to the first efforts to settle the island in the 9th century.
Ingólfur Arnarson settled with his family around 874 at a place he called Bay of Smokes which is now known as Reykjavík. He arrived following Flóki Vilgerðarson who named the island Ísland in the 9th century.
Genetic evidence suggests that approximately 62% of the Icelandic maternal gene pool is derived from Ireland and Scotland. The Irish and Scottish Gaels were either slaves or servants of the Norse chiefs according to the Icelandic sagas.
The chieftains met on the Þingvellir plain near Reykjavík in 930 to establish the Alþingi as Iceland's first national parliament. In 1798 the Alþingi was discontinued for several decades before being restored in 1844.
On the 17th of June 1944 the monarchy was abolished and a republic was established on Jón Sigurðsson's 133rd birthday. This event ended nearly six centuries of ties with Denmark after Iceland gained full sovereignty in 1918.