Short answers, pulled from the story.
Domicile serves as a legal anchor for an individual's personal law covering status and property rights regardless of nationality. Unlike nationality which ties a person to a country, domicile focuses on where one lives with future intent.
Udny's domicile of origin at birth was Scotland even though he was born and lived in Tuscany while his father resided in England. The court determined this based on the fact that Udny's father's domicile of origin was also Scottish.
Canada's Divorce Act of 1968 replaced domicile as the basis for provincial jurisdiction over divorce proceedings. By March 2021 this rule remained in effect across Canadian provinces allowing courts to exercise jurisdiction if either spouse has been habitually resident in the province for at least one year immediately preceding the commencement of the proceeding.
Commercial domicile appears in public international law during times of war specifically within prize law where a merchant's status as enemy or neutral is determined in courts of a belligerent state based on where they reside and carry on business without intending to end trade immediately. Any property belonging to a merchant of a belligerent state is liable to capture at sea if they conduct business there.
Section 267(1)(b) of the Inheritance Tax Act 1984 defines a fifteen-year rule for deemed UK domicile status requiring a person treated as domiciled in the UK to have been resident there for at least fifteen of the twenty tax years immediately preceding the relevant tax year.