My guess is that once the title reverts to the crown upon Andrew's death, it won't be recreated again until Louis reaches his majority and gets married. He does hold the spot of the second son when William becomes the monarch.
King James II was created Duke of York as the second son of Charles I and, as king, was later deposed in the revolution of 1688 for his alleged absolutist tendencies in addition to his Catholic faith. That did not prevent the title, however, from being recreated later for other second-born sons of British monarchs, including the future Kings George V and George VI, and now Prince Andrew. I believe the tradition will be maintained regardless of the veredict in Andrew's lawsuit, but, in any case, whether Andrew keeps the dukedom or it is forfeited, it will not be recreated as long as Andrew is still alive, and he may well be by the time Prince Louis (the natural next candidate to the title) gets married.
If the norms of lifespan and marriage in the current BRF hold for Andrew and Louis respectively, it is quite likely that Andrew will be alive when Louis marries.
I find it interesting that although many BRF observers regard the family as highly valuing tradition, it seems to be universally taken for granted that Elizabeth's nontraditional choice to wait until princes' marriages to create peerages for them will be continued in future reigns and become a new tradition.
Personally, I hope future monarchs will revert to the old tradition, or even better, introduce a new precedent of creating royal peerages either at birth or on coming of age. In today's media climate, I think that no matter the correct protocol, it is unrealistic to expect the public to adopt a completely new style for a royal after he has lived for decades in the public eye. The Queen's elderly cousins are normally correctly called the Duke of Gloucester and the Duke of Kent by the public and in the media, but her sons are commonly called Prince Andrew and Prince Edward as much as Duke of York and Earl of Wessex, and her grandsons are evidently better known to the general public as Prince William and Prince Harry than as Duke of Cambridge and Duke of Sussex, since both have used Prince instead of Duke in international contexts for the name recognition.
The MP for York Central and the member of the York City Council should know that the Duke cannot legally "relinquish" his peerage. As far as I understand, people who succeed to a hereditary peerage may, under the Peerage Act 1963, disclaim the peerage within one year of succeeding thereto. I don't think the aforementioned act applies, however, to newly created peerages such as the Duke's and, in any case, Prince Andrew has been the Duke of York since 1986, so the act could not apply to him anyway.
That is quite true. But I am not sure the two politicians are necessarily ignorant of the law. In law, the dukedom cannot be unilaterally relinquished, but if the Duke petitioned Parliament to act on his hypothetical will to relinquish, I suspect his request would be accepted without too much fuss. In contrast, a bill to strip the dukedom without his consent would probably be much more controversial (not so much out of concern for Andrew but because of the precedents it would establish).
Traditionally they didn't precede their younger brothers in the line of succession.That has changed so the tradition of giving peerage titles to sons but not daughters should change accordingly, just as the BRF as abandoned other outdated traditions (arranged marriages, for example).
I wholeheartedly agree. Unfortunately, there is the legitimate argument that, since all British governments thus far have clung to maintaining a (nearly) male-only hereditary peerage, the monarch granting hereditary peerages to female members of the royal family could easily be perceived as taking a political stance.
However, I see nothing stopping the BRF from abandoning some other outdated traditions (e.g., the insistence on addressing women by their husbands' first and last names whether they like it or not).