Short answers, pulled from the story.
The Danish Realm, officially the Kingdom of Denmark, is a sovereign state made up of three constituent territories: metropolitan Denmark, the Faroe Islands in the North Atlantic, and Greenland in North America. The territories are united under the Constitutional Act of Denmark and the concept known as Rigsfællesskabet, meaning the unity of the Realm.
The Faroe Islands and Greenland have been under the Crown of Denmark since 1397, when the Kalmar Union was ratified. They became part of the Danish Realm in a de jure sense in 1814, when Denmark kept control of both territories after ceding Norway to Sweden under the Treaty of Kiel.
Greenland gained self-rule on the 21st of June 2009, following a referendum held on the 25th of November 2008. Self-rule replaced the previous home rule arrangement established in 1979 and gave Greenland control over judicial affairs, policing, natural resources, immigration and border controls.
Both territories remain outside the European Union primarily due to disagreements over the Common Fisheries Policy. The Faroe Islands were never part of the EU, as explicitly stated in the Rome treaties. Greenland joined the European Communities as part of Denmark in 1973 but opted to leave in 1985 after gaining home rule in 1979.
The entire Danish Realm covers an area of 2.2 million square kilometres, ranking it as the twelfth-largest country in the world according to The World Factbook. Greenland alone accounts for 98% of that area and holds the same ranking by itself. Denmark proper covers about 43,000 km2, ranking 133rd on the same list.
Yes, the 2009 self-rule act established a formal pathway to Greenlandic independence. The Greenlandic people must first decide to pursue independence, after which the Naalakkersuisut and the Danish government negotiate the terms. The resulting agreement must be ratified by the Greenlandic Inatsisartut, approved in a Greenlandic referendum, and receive consent from the Danish Folketing under section 19 of the Danish constitution.