Short answers, pulled from the story.
The London Gazette of the 4th of June 1814 refers to him having the title of Duke of Wellington. The titles were bestowed upon him on the 3rd of May 1814 after he returned from Spain.
The Prince Regent created the title on behalf of George III. Richard Wellesley, Earl of Mornington, discovered a manor in the parish of Wellington, Somerset, which served as the source for the dukedom name.
The Dukes of Wellington hold the titles of Prince of Waterloo of the Kingdom of the Netherlands and Duke of Ciudad Rodrigo of the Kingdom of Spain with Grandeeship. These foreign peerages were granted as victory titles for distinguished service abroad during the Peninsular War.
Arthur Richard Wellesley succeeded his father in 1852 and held the title until 1884. Henry Wellesley followed him from 1884 to 1900 as the nephew of the preceding holder, and Arthur Charles Wellesley held the position from 1900 to 1934 as the brother of the preceding Duke.
The family seat is Stratfield Saye House near Basingstoke, Hampshire. This property has been the home since the first Duke's acquisition, while Apsley House in London is now owned by English Heritage.