The chimpanzee Coco that Frasier suspects is the murderer is stated as being Helen's pet. Daphne even shows Frasier a picture of Helen together "with her pet monkey Coco." However, the theory that Frasier presents has Coco sneaking into and out of Helen's house, which would not be necessary if it were her own pet.
Martin is often saying that it's taken him 20 years to solve the murder. But when he's sitting at the table with his police buddies, he mentions that it's taken him 30 years and that it took Frasier "with his PHD mind to crack this baby."
The chimpanzee who Frasier suspected of committing the Weeping Lotus murder was referred to throughout the episode as a monkey. Chimpanzees are apes, not monkeys.
Even an actual retired police detective would not be allowed to take an unsolved case file home.
Frasier describes "Inherit the Wind" as a Pulitzer Prize-winning novel. It is a play, not a novel, and did not win the Pulitzer. Furthermore, Frasier mispronounces Pulitzer.