William, Harry and Kate's real surname (and the House they all belong to) is Windsor.
Are sure about that, Artemisia? I have always tought that, traditionally, Royals have no surnames and only given names. The names of their royal houses or countries or peerages are not their legal surnames but obviously that's what identifies them. Their sexes are recorded in their birth certificates as Princes or Princesses, not males or females or boys or girls. Thus, the Duke of Cambridge is Prince William, Arthur, Philip, Louis, Duke of Cambridge, etc., etc. Kate is not Catherine Windsor but Princess William, Arthur, etc., etc. Those more remote members of the Royal House who are not Princes and Princesses need surnames, of course, and then it is Windsor and Mountbatten-Windsor for male-line descendants of Elizabeth II. It is not a surname passed from father to son as it is usual in common families but regulated by the Letters Patent regarding the descendants of the Royal House.
Correct. Any non HRH children will be Lord or Lady Christian name Mountbatten Windsor without the "of Cambridge". Only the one who is an HRH will use the designation "of Cambridge".
Only the first son of Prince William will be a Prince and his designation will be of Cambridge. Any further children, boys and girls, will be just Lord and Ladies and thus, with the name of Mountbatten-Windsor as surname.
Well, they will all be one day children of the souverain, so it's really not that big an issue. I'm more interested in the fact that the Letters Patent for William again reduces the title and succession to "heirs male" which is traditional and not gender equal.
In 1900, Queen Victoria created her granddaughter's husband, Alexander, the Duke of Fife, a second Duke of Fife, this time in the Peerage of the United Kingdom and with a special remainder that allowed the title to pass to his daughters and then to the heirs male of their bodies. That was needed because Princess Louise, future Edward VII's eldest daughter and future Princess Royal, had only two daughters then after 11 years of marriage. She was 33 years old then and there was a real possibility that she would not give birth to a male heir. William and Catherine are not in the same situation so the Queen did it in the traditional way and the Letters Patent contains the standard remainder. It is not important any way as he is King-to-be and the Dukedom is going to revert to the Crown upon his accession.
An another example of unusual remainder in a grant of peerage is the Duke of Marlborough's. For more info see here
http://en.wikipedia.org/wiki/Duke_of_Marlborough_%28title%29
That was also a case when a really powerful peer had no legal male heirs of his body so it was decided that his title can pass through a different line.
I think they need to abolish the male preference primogeniture in the order of succession to the throne first, and then there may be some changes in the Peerage system.