What makes cables suitable or unsuitable for a given purpose?
Your main considerations should be as follows for your cable:


Flex life, i.e. how long the cable can be flexed before it breaks
(very important for your application)
Shielding/grounding for the composite video portion and other low-level signals 
Impedance for signals with content above 16 MHz (see calculation below)


For a cable run of 4 feet and assuming a typical velocity of propagation of ~66% for the cable, the one-tenth wavelength rule says you can ignore transmission line effects below about 16 MHz. I don't know enough about the frequency content of your I2C lines to give a recommendation there, but analog baseband composite video will have frequency content below 8 MHz in all regions of the world.

If you decide to go with a Cat 5E Ethernet product, try to find one designed for continuous flexing. Those types of products are used in industrial robotics products.

For the composite video portion, you have two choices:


Make a composite cable (cable made up of multiple elements) yourself with a coaxial element and your Ethernet element.
Use a 100-ohm twisted pair from the Ethernet with a balun.


If you decide to make a composite cable yourself, find a 75-ohm video product designed for continuous flexing (long flex life). Combine it with the Ethernet element in a braid.

To maximize your flexibility and the flex life of your finished product, you want to use flexible materials (soft plastic and many strands of copper) and construct things in such a way as to minimize the twisting and untwisting of the internal elements.

I'm a former applications engineer for a wire and cable company. Here is a link to a general-audience presentation about wire and cable: http://www.belden.com/resourcecenter/cablebasics/upload/Cable-101.pdf

And here is a link to industrial products in a wire and cable catalog: https://www.belden.com/docs/upload/Cabling-Solutions-for-Industrial-Applications.pdf

This 2002 book is also a good overview of the audio/video cable industry, even if it is somewhat outdated:
http://www.amazon.com/Audio-Video-Installers-Pocket-Reference/dp/0071386211