What is the relationship between ARM® and TI OMAP® / Samsung Exynos® / Apple A5X® / Qualcomm Snapdragon® / Nvidia Tegra®?
1ARM is a processor core alone - it consists of something like Verilog or VHDL that describes the logic of the processor core.  While ARM designs the processor, they don't make chips - you can't go the ARM website, click on 'shop' and get a brand new ARM microcontroller chip - it just doesn't exist. 

1) ARM sells its processor designs to other companies.  Those companies take processor design and integrate it with a whole buncha other components onto one System On Chip.  So each of the manufacturers you mentioned have chips with ARM cores, but different peripherals.  Apple doesn't make its living selling microcontrollers - they integrate every possible peripheral (processor, display adapter, memory, ethernet controller, you name it!) on to one piece of silicon and put it in an iPad, iPhone, etc.  Companies like TI make general-purpose processors to fill a variety of niches - all with ARM processor cores and different peripherals.  Then they sell it direct to you.  

2 )The business relationships probably involve large amounts of cash from companies like TI and Apple to ARM, with processor designs coming from ARM to the other companies.

3) Widely varies.  There's no graphics processor compatible with a TI Stellaris LM3S6432 - it's a Cortex M3-based microcontroller.  Other, more capable processors probably have a wide variety of compatibility, but there's so many ARM chips out there there's no sense in me even trying to compile a list.