Short answers, pulled from the story.
The Crown is a legal entity that exists as a corporation sole transcending the mortal body of the monarch. It functions as the state, the government, and the monarch simultaneously while remaining distinct from all three. This unique legal personality holds all executive, legislative, and judicial powers in a single indivisible package.
The term the Crown was first defined as an imperial crown during the reign of Henry VIII in the Ecclesiastical Appeals Act 1532. This declaration asserted that the realm of England is an empire governed by one Supreme Head and King having the dignity and royal estate of the imperial Crown of the same.
In Canada the Crown is the prosecuting party led by a Crown prosecutor or Crown attorney and cases are designated as R v defendant. Most jurisdictions in Australia use R or The King or The Queen in criminal cases while Western Australia and Tasmania bring prosecutions in the name of the respective state instead of the Crown.
The Statute of Westminster 1931 enshrined the unity of action in law by establishing a constitutional convention requiring the assent of each dominion parliament for alterations to the Succession to the Throne. This statute ensured that the Crown in right of each realm and territory acts independently of the other realms and territories.
The judgement in Ex parte Indian Association of Alberta ruled that the obligations of the Crown towards the indigenous peoples in Canada were held by the Crown in right of Canada and not the Crown in right of the United Kingdom. Eugene Forsey wrote that the Queen is Queen of Canada and not Queen of Ontario Queen of Quebec or Queen of British Columbia.
A heraldic St Edward's Crown was restored in 1952 when the reign of Elizabeth II began. The Tudor Crown design had been standardised in 1901 and continued in use until that restoration.