In the UK he wouldn't be introduced like this; he would be HRH The Duke of Sussex; they wouldn't combine the 'prince Harry' and 'Duke of Sussex' (without 'the'!). Given that he is not supposed to use HRH; using his highest title would mean being introduced as 'The Duke of Sussex' without a first name or prince; however, I guess both Harry and Meghan prefer to have their first name attached to it for recognition purposes.
In official documents in the UK (probably in his British passport), he is most likely cited as His Royal Highness Prince Henry Charles Albert David, Duke of Sussex.
As I argued before, the difference between the style above and HRH The Duke of Sussex is simply a matter of long versus short style, but the longer form is perfectly valid in the UK and, I insist, is actually the form used in writing in official documents.
I agree, however, that, in the UK, he would be introduced, e.g. in a public event like a concert, as HRH The Duke of Sussex, and not as Prince Harry, Duke of Sussex.
Prince Harry, Duke of Sussex and Meghan, The Duchess of Sussex seem to be the "brand names" the couple adopted in the US, which makes a lot of sense since many Americans might not recognize them solely by their titles of Duke/Duchess of Sussex, but will recognize them as Prince Harry and Meghan (Markle).
The USA no longer requires you to renounce citizenship when becoming a citizen. I acquired US citizenship a number of years ago. (I have an American mother and a Canadian dad, born in Canada. I'm a dual Canadian/American citizen). I have to swear a brief oath. My mom is the opposite of me (American citizen originally, acquired Canadian citizenship, now dual citizen.)
The only sticking point for Harry would be his British titles (the US does require renounciation of foreign titles).
Yes, you are correct. The naturalization oath only requires that you renounce
allegiance to your country of origin, but not that you renounce your other citizenship properly.
I wonder how "allegiance" is measured in practice however. Some countries seem to follow a legal doctrine that citizenship automatically implies an obligation of allegiance to the Sovereign, so I fail to see how one could in practice renounce allegiance in those cases without explicitly renouncing citizenship too. Does anybody know?