Working on the assumption most people aren't reading kid's puzzle books (although it is a pretty fun, more relaxed time if you fancy it!), I've compiled a list of main series related observations. Please only read if you don't mind spoilers - and the kids aren't mentioned much since they are Jr originals.
The book seems to take place between books 1 and 2 (plausibly could be during book 1, but unlikely)
Many of the events of book 2 and 3 were in motion beforehand, especially with Tekco
Rose, Raspberry and Indigo all have kids - Indigo's specifically is Olivia, one of the main detectives
This means Logico helped murder his junior employee's dad during book 3 (ouch)
Applegreen, Lapis, Saffron and Ruby work in the same school, alongside Raspberry Jr
Emerald and Earl Grey are neighbours
Logico seems to be a bit famous now, at least enough for strangers to recognise him and to be signing stuff
The Detective club is really well funded, and Logico appears to be in a high-up role, if not the president
Irratino is a "pretty swell" Uncle (likely a second cousin rather than secret sibling)
Irratino is consequently pretty good with kids
Logico is TERRIBLE with kids
Irratino has multiple classic secret levers and switches installed for secret rooms and passages in the II (something we'd all probably do with comical amounts of money to be fair)
Irratino has great memory, and has the entire Institute memorised (which may also be how he can navigate the impossible maze with little trouble)
While it's no surprise, Irratino is confirmed as a habitual wonderer and prone to boredom (AuDHD king)
Irratino is similar to Logico in that he likes being included as a suspect when relevant
Logico either does a decent amount of undercover work, or has a LOT of side jobs
Logico earns what I believe is his first colour association - marzipan! (Suitably beige in my opinion)
At least two of the four detectives don't recognise Logico with a very mild change in appearance (mustache and hat) but clearly know who he is - either face blindness is super common in Murdle, or Logico is gathering an army of specifically autistic children
Logico has a houseboat (unclear if it's a full-time home or for the case - I like to imagine it is indeed his)
Logico is good with computers - at least good enough to stump a tech prodigy
Logico is kinda bad at tone/jokes
The DC and II seem to be at least a little linked together now - Logico tries to recruit two of the kids at the II, and Irratino helps with a DC training exercise as a "favour"
The II has at least one other branch, on the Violet Isles
A funny possible plot hole - Logico knew at least part of SPY was corrupt before the events of book 3 (between that and SoM, I'm starting to think Logico got hit a bit too hard during chessboxing - that or I'm checking timelines in a children's book)
Logico not only successfully pulls off the patented Tino-death-trick, but Irratino (presumably) fails to pull off the same trick
Logico is, and I cannot overstate this, the dopiest, most awkward motherfucker on the planet. Seriously, it's so funny seeing him without his internal dialogue colouring all his actions. It's like if Columbo was actually Like That. He thinks he's super cool and awesome and he's actually Laios from dunmeshi (AuDHD king). He's smart in the logic department but the rest of his brain is empty.
NO BUT SERIOUSLY. He earnestly does the Perry the Platypus disguise TWICE. He wears his hat under a second hat in case he needs to do a dramatic reveal. He lets children go to a wartorn country and solve several murders with no plan while believing that this is a totally planned excursion. HE HIRES A CAT. HE LETS A SMALL CHILD BELIEVE HE IS GONNA BE A DETECTIVE BEFORE GOING "oh no sorry that's for the cat, welcome to the force Mr McPaws" WITH GENUINE SERIOUS INTENTIONS.
If I'm honest, I think the cat might be the smartest detective there (including the adults)
