the throne of Ankh-Morpork DOES physically exist. Vetinari sits next to it in Guards! Guards! much in the fashion of Tolkien's Steward of Gondor. It may be too rotten to sit on, but it is generally understood to physically exist and perhaps more importantly, to represent the legitimacy of the State.
Likewise there is an attempt at one point, to present Nobby Nobbs as a front for an attempt to usurp power from Vetinari on the basis of his supposed nobility.
Vetinari may present himself as having no royal or aristocratic connections, but he appoints Vimes a Duke because he finds it useful to do so, and everyone accepts this as a fact.
Nobility is still important in Ankh-Morpork; as further evidence, Lord Rust's military ventures in Jingo. This is also a round-world trait; look at King Juan Carlos of Spain's role in supressing an attempted military coup, or the fact that the German Ambassador to the Court of St James still calls himself a "Baron".
Carrot's main role as the Heir Apparent may be of little practical value in the sense that he is unlikely to be King, but while he is generally known to exist and accepted as the heir apparent, it does prevent anyone else from claiming the role.
I wouldn't be too dogmatic about the A-M timeline. I'd always had the impression that Carrot arrived in A-M as a naive young man, possibly in his late teens. The more mature Carrot who develops a relationship with Angua has moved on quite some way from this. I'd accept that the stories cover a period of less than ten years, but they are inconsistent in detail and after all, carrot is well-established in A-M when Angua arrives. I'd also rather had the impression that she was somewhat older, or at least more adult, in the early stages of their relationship.
Vimes, after all, has time to get a grip on his alcoholism, expand the watch, preside over the training of officers who move to various other cities, meet Sybil, marry and have a son who is old enough to be read stories and be aware of expecting this. On that basis I'd say the Vimes story arc covers a period of five to ten years, and nearer ten. The various background detail of Thud! and Nightwatch make it fairly clear that he is at that point, in his early 40s ( since he is about 18 in the flashback section of Nightwatch, stated as taking place twenty-five years before ) and has been married not less than three years.
Nightwatch makes it fairly clear that Colon is an experienced and established Watchman at the time Vimes joined, hence he is probably in his mid-fifties at the time of Thud! with Nobby being somewhat younger - there is at least ten years between the two men, probably more, per Nightwatch.
On that basis, Carrot and Angua's relationship probably spans a period of not less than five years, since he arrives before she does.
One thing I'm unclear about, is Colon's and Nobbs' military experience. Colon appears to have been in the army as a young man. He may have become a Watch sergeant having been a sergeant in the army. However the young Nobbs of Nightwatch is clearly too young to have served, especially as he apppears to have been known to Colon during this time, which leads to the conclusion that Nobby and Fred served together at some time between the two periods of Nightwatch
anyone?