On 26 May 1342 he inherited the vast Douglas family estates from his uncle, Hugh Douglas.1 In August 1353 he slew his kinsman, William Douglas.1 He fought in the Battle of Poitiers on 19 September 1356, where he was wounded.1 As a result of his marriage, William Douglas, 1st Earl of Douglas was styled as Earl of Mar before 13 November 1357. He was created 1st Earl of Douglas [Scotland] on 26 January 1357/58.2 He held the office of Warden of the Marches.
Created 1st Earl of Douglas in 1358, and by marriage Earl of Mar. He returned from training in France in 1348 and recovered Douglasdale and Ettrick Forest to Scottish allegience. He was present at the Battle of Poitiers in 1356. He was warden of the east marches. He rebelled against King David II in 1363 in consequence of royal misappropriation of funds raised for ransom. He was reconciled and swore allegiance to Robert II in 1371. He was justiciar of southern Scotland from 1371 until his death. He had issue by his wife's sister-in-law, Margaret, daughter of Thomas Stewart, Earl of Angus, and Margaret St. Clair.
Sir James Balfour Paul, Editor, 1906, The Scots Peerage, writes: "Hume of. Godscroft and others have assigned three wives to this Earl of Douglas: first, Margaret, or Agnes, Dunbar,2 who is said to be the mother of James, second Earl of Douglas, and of Archibald Douglas, Lord of Galloway; second, Margaret of Mar ; and third, Margaret Stewart, Countess of Mar and Angus. But his only wife was Margaret of Mar, daughter of Donald, and sister of Thomas, Earl of Mar. Douglas and she were married, so far as can be ascertained, in 1357, 3 and she survived him, marrying, as her second husband, before July 1388, Sir John Swinton of Swinton, and dying in 1390."
Note: Y-DNA Haplogroup conflict. (Probably does not belong to Haplogroup I2b1, as listed on familytreedna.com mrl
William, 1st Earl of Douglas, incl Douglas-Hamilton, Dukes of Hamilton, I2b1 Group Type 1: William, 1st Earl of Douglas, incl Douglas-Hamilton, Dukes of Hamilton, Y-DNA Haplogroup I2b1familytreedna
"Royal Ancestry" 2013 Douglas Richardson Vol. III. p. 151
Johnston, George Harvey. "The Earls of Douglas." The Heraldry of the Douglases: With Notes on All the Males of the Family, Descriptions of the Arms, Plates and Pedigrees. Edinburgh: W. & A.K. Johnston, Limited, 1907. 14-15. Print.
It may be possible to confirm family relationships with William by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree:
I am not sure this still an active question. Eugene is correct here. William did marry Margaret of Mar. She was the sister of Thomas of Mar, the last of the Mormaers of Mar. When Thomas died in 1377 the estates of Mar passed to Margaret and thus to William and he became Earl of Mar before that of Douglas. This is mentioned here. However William also had an affair with Thomas' widow, Margaret Stewart, Countess of Angus and had certainly a son - George who inherits the Angus estates and likely a daughter Margaret who receives lands from her sister Isabel after Isabel becomes Countess of Mar. Most of this is covered on wikipedia.