Mrs. Slocombe had multiple personality disorder.
Mrs. Slocombe is an Auror.That's why her hair changes color.
Mr. Humphries is Mr. Grainger's illegitimate son (or at least Mr. Grainger thinks he is).Remember when Mr. Grainger mentioned that he once "got a girl in trouble"? Nothing more is said about this during his time on the show, but later we had an episode with Mr. Humphries thinking that Old Mr. Grace was his father and his mother basically admitting that she didn't know who his father really was. Given that his mother was the one who helped Humphries get the job with the firm in the first place, it's also likely that she prevailed upon Mr. Grainger to keep an eye on "their boy" throughout his career. And Grainger shows FAR more affection for Humphries than anyone else on the floor, and seems to take his quirks and affectations in stride. Since Lloyd and Croft used the same plot contrivance on Dad's Army (with Sgt. Wilson and Pike), it's safe to assume that this idea would have been explored more thoroughly if Arthur Brough hadn't passed on after Series Five.