4 player splitscreen shooters
America's Army
Call of Duty 3
Call of Duty 4
Call of Duty: World at War
Conflict: Denied Ops
Dark Sector
Doom (XBLA)
Far Cry Instincts Predator
Halo 3
Halo 3: ODST
Marathon: Durandel (XBLA)
Perfect Dark Zero
The Club
The Darkness
Tom Clancy's Ghost Recon Advanced Warfighter
Tom Clancy's Ghost Recon Advanced Warfighter 2
I don't know if any of these have bots. Unreal Tournament 3 has bots, but it only has 2 player offline splitscreen to go along with them. The only games I've heard have bots are Perfect Dark Zero, Gears of War 2, the upcoming Perfect Dark remake for XBLA and Section 8. Turok might also have bots because it has 2 player split screen, and that just sounds remarkably dull without bots to go along with it.
Regarding racers, 4 player split screen games are as follows:
Baja: Edge of Control
Blur (Coming out November 3rd)
Mad Tracks (XBLA)
Moto GP '06
Moto GP '07
SBK Superbike World Championship
Sonic and SEGA All-Stars Racing (Coming in 2010)
Split/Second (Coming in 2010)
Stuntman Ignition
Vigilante 8 Arcade (XBLA)
Hope this helps.