There were no ducal titles in the 13th century in either England or Scotland.

The title of Duke in the British Isles was not used until about 1330 in England (1337 is the date of the creation of the Duchy of Cornwall).

On the main continent of Europe ("the Continent"), the ducal titles previously (by about the 11th century at latest) meant a ruler who was nearly sovereign within their lands, but still owed homage to a king for their territory. (Consider the Duke of Normandy, the Duke of Aquitaine, etc).

After King John of England (died 1216) lost the vast majority of his family's inherited lands on the Continent, the title of "duke" was regranted as an appanage (to the younger children of the King of France), but no longer carried the same connotation of near-sovereignty that it had before.

It did, however, have a connotation of rank in the peerage that outstripped Counts (the Continental equivalent to English and Scottish Earls). In 1337, King Edward III of England elevated the title of Earl of Cornwall from an Earldom to a Dukedom (possibly because Edward III had Continental ambitions and wanted to compete on the same level as the French aristocracy). While Scotland had close diplomatic ties to France (due to the fairly new at the time "Auld Alliance" between France and Scotland), it tended to follow England in terms of royal precedents. The Dukedom of Albany was not created until nearly two centuries after the 13th century, in 1398 for the brother of King Robert III of Scotland.

Most of the earliest Dukedoms in both England and Scotland remained in the royal family (much as today new creations of Dukedoms in the Peerage of the United Kingdom are almost universally for direct descendants of Queen Elizabeth II). (Side note: One of the earliest Dukedoms in England, the Duchy of Lancaster, was not created for a son of King Edward III. It was created for a close royal relative, however, to honor his achievements on the battlefield. Since Henry of Grosmont, Duke of Lancaster only had daughters, it was destined to become extinct on Duke Henry's death rather than create a noble family that might compete with the royal family for prestige. Then Edward III realized he could marry off his third (surviving) son John of Gaunt to the fabulously wealthy heiress of the first Duke of Lancaster and re-create the title for John, and history was off and running.)

The greatest titles in both England and Scotland prior to the invention of English and Scottish Dukedoms were Earldoms and in practice Earls were the highest members of noble society. As such, their households would be in keeping with a royal standard, although not quite as elaborate as the King's household.

A royal or near-royal Scottish Earl would very likely have a fairly standard household staff, although probably not as extensive as in other European countries. There would probably be a Steward of the Household (the Steward of the Royal Household, by the way, is the origin of the Stewart/Stuart dynasty on the Scottish and English thrones). They would likely have a clerk to take written notation and a confessor as their personal religious advisor. They might have a Chamberlin, although the line starts getting blurred a bit between what a Steward's responsibilities would be and what a Chamberlin's responsibilities would be.