The identity morphism is a path of length zero! For some reason I never tried to see it that way. Since all composites exist (say, because it's easy to check – proving that two paths commute is the hard part) I always thought of all links (conflating paths and arrows) as essentially having length 1. Which sort of works, too, but having paths and arrows separate like this is nice!
Regarding puzzle 104, hah, there's just one canonically famous sequence... However, I had to look twice to get rid of the feeling that things should get multiplied somehow.
It seems we should call these categories Fibonacci and Pell. Is that something people do?