If, say, the Duke of Grafton had married the then Princess Elizabeth he may have been given his own Dukedom on his marriage (like the DoE), as at the time he wasn't the Duke, just his father's heir apparent and held no titles in his own right. He may have also then been created a HRH and a Prince, or he may not have - the DoE was styled as an HRH right away, but wasn't created a Prince in the UK until well into his marriage.
Had she married an Earl (or another peer lower than a Duke) it's likely he would have been created a Duke, like as what happened when Princess Louise, daughter of the future Edward VII, married the then Earl of Fife.
The title situation wouldn't be too dissimilar from the situation we have now; their eldest son would be in line to inherit both their titles, and upon doing so his father's titles would merge with the crown. It's possible that if a woman inherited the crown without the peerage being merged with the crown then the peerage would pass on to someone else - so, like now, if something happened to Charles, William, and George, causing Charlotte to be the Queen's heir apparent then the DoE title would go to Andrew and not merge with the crown at any point. It's also possible that if the Queen and her husband had more than one son, or if the heir apparent had more than one son, they could decide to have the Dukedom recreate for a younger son after it merges with the crown - which is what is planned with the DoE's title now; it's expected to be recreated for Edward after it merges with the crown.
As for properties... They would be inherited by any children of the Queen and her husband, but they would be private property (like Sandringham), not Crown property (like Buckingham Palace).