A Taste of Power: The E3 demo had the majority of the President's super powers unlocked from the get-go, so people sampling the demo could whet their appetite for when the game is released and they need to progress through the game to unlock their powers.
Bad Export for You: The Australian version, due to the removal of Shaundi's loyalty mission.
Banned In Australia: Two attempts to submit the game were rejected because of the alien probe gun as well as one of Shaundi's missions that involved drugs. Surprisingly the anal rape was allowed to stay as DLC, it was the drugs that drove Australian censors up the wall and once that was removed the game was allowed, even rated MA rather than the new R rating.
Dueling Games: With Grand Theft Auto V, which released about a month after Saints Row IV. Despite Saints getting positive reception and good sales, GTA blew it out of the water, in both reviews and sales.
Lin, Donnie, William Sharp, Dane Vogel, Viola, Loren, and Zimos were originally going to be homies in game and had cut audio logs. Zimos' clothes are available for purchase in stores, and his personal symbol shows up on the Saints' mothership, so it is possible they may return in DLC episodes. Zimos has indeed returned in Enter the Dominatrix.
The "aliens trap the Saints in a virtual Steelport" aspect was originally going to be a stand-alone expansion for The Third, called Enter the Dominatrix. It was later decided to cancel the expansion and just merge that and the concept for IV together. Who knows what could have happened if the "virtual Steelport" and "the Boss as the President" plots stayed separate.
Later on, Enter The Dominatrix was made a DLC pack, and several previous characters do indeed return, answering both of the above points.