During the series run, Florence Henderson lobbied the producers constantly to allow Carol Brady to get out into the workforce. Henderson thought this would be more in line with how she was in real life. The producers kept the character of Carol Brady unemployed, though she frequently did volunteer work and fundraising for charity.
A scene in The Honeymoon (1969) makes it clear Mike's first wife had died, making him a widower, but the status of Carol's first marriage was kept a secret. Creator Sherwood Schwartz maintained Carol was divorced from her first husband, but nothing about it was mentioned on the series. At that time, divorce was a subject matter that was still considered largely taboo for television, particularly a series aimed at family audiences.
The sliding glass door in the back of the Brady house actually never had glass in it. It was simply put on the sliders with no glass so as to prevent glares from lighting while filming.
Robert Reed was a closeted gay man. Florence Henderson noticed early on that he appeared uncomfortable acting romantically with her, and he admitted the truth to her privately. They worked around it, rehearsing kissing and hugging scenes off-camera, so they would look more natural when it came time to film them. The entire cast kept Reed's sexual orientation a secret until after his death.
The family dog "Tiger" was killed by a car in season one of the show before the filming of episode five was completed. A replacement dog proved to be unworkable. Tiger's doghouse remained on the set, though, because one of the studio lights fell and burned a hole through the astroturf, and the doghouse was used to hide the burned spot.