My best interpretation of the sketch is that it is Vimes, apparently at the wheel of a side-wheel river steamer in heavy water. Maybe it's a Riverworld joke? Suggests a Genua location since these are specifically mentioned in that context elsewhere.
I suggest the creature on his shoulder is a gargoyle, but I have no suggestions regarding the chickens
regarding Carrot and Angua, surely the fact that the books are published more-or-less annually but books in a given story arc aren't published consecutively, is pretty much irrelevant? Discworld elapsed time is surely the sum of identifiable events in the various story arcs. Some are approximately contemporary and some are clearly out-of-sequence, especially Small Gods and to a fair extent, Pyramids.
Broadly speaking I'd say the books can be divided into several periods;
1) out-of-sequence stories like Small Gods, and arguably Pyramids with little or no identifiable connection to the main arcs. Early books like Colour of Magic, Sourcery and Light Fantastic belong here to varying extents, because they don't form a coherent part of the canon as it subsequently develops.
2) flashback stories like the early sections of Nightwatch, or the sections relating to Granny Weather wax and Mustrum Ridcully's younger days
3) the main story arc which involves the main characters
I'd say the main story arc spans a period of twelve to fifteen years. This would make Carrot somewhere in his late teens when he arrives in Ank-Morpork, with Angua about the same or slightly older. Vimes would be somewhere in his thirties when he first appears in Guards! Guards! and late 40s in the later books. Fred Colon would be early 40s when he first appears in Nightwatch, and approaching sixty in later books.
The senior witches would be somewhere between their late forties to early sixties; it's clear that some witches live to a great age but Granny and Nanny are too active for this, and anyway in a society like Lancre, Nanny might well be a grandmother in her late thirties. Her ( aparently ) eldest son ( the blacksmith ) would then be somewhere between early twenties and mid-to-late thirties. So is Magrat, with Verence again somewhat older.
To some extent the DW books display the same dislocations as the Aubrey/Maturin novels. Aubrey is a young lieutenant when he first appears, and his career runs in approximately real time ( as identifiable from real-history events ) for the next five or six books. The last two books run in real time, and his age is approximately correct relative to the earlier ones, as are his family.
However the middle part of the sequence comprise a series of stories broadly related to real-world events. They are internally consistent, but add up to a considerably longer period than can be accomodated between the fixed opening and closing sequences.
There are also various non-sequiturs which are essentially driven by plot requirements. Midshipman Read apparently loses an arm at the shoulder, but subsequently re-appears with a hook - because the plot requires a junior officer well-known to Aubrey for a specific role. Aubrey has a long-serving officer with a wooden leg on one occasion - appointing hin captain of a prize as a means of pensioning him off honourably - whereas another fellow officer is described as having moved to the role of captain of an East Indiaman, due to a wooden leg, because the plot requires an officer known to Aubrey, witha naval background but retired from the service with no reflection upon his competence.
Carrot and Angua are in this category. They are favourites with the readers and can't be dropped, but they can't simply decline with age as Colon does. Vimes visibly ages and grows into his responsibilities. The Patrician can remain ageless, but he does appear to age since he walks with a cane in later appearances. Detritus develops greatly over time.
Logically, Carrot and Angua should have become minor background characters but the Watch stories require someone in their roles. Cheery isn't it, and nor is Detritus.
I suspect that Carrot and Angua's relationship is a piece of misdirection, covering the fact the TP hasn't resolved it himself.