Is there any NEC code requirement for how far off the floor Category 5 wiring must be run?
No.  Cat 5 cabling can be run on the ground or underground.  If you are worried about it working just make sure the coating isn't torn/cut and it is capped properly.  If you are worried about code violations - no inspector cares unless the cable is interfering with something else he cares about.

My experience level - ran communications cable for large internet provider for 4 years both in the field and in-house.  If you run cable underground there is code specific sheathing needed.  Not really sure this would apply for cat5 because that wouldn't be used for long runs.  I personally have buried cat5 cable 1 foot under by 50 feet to set up a link in a shed that now works just fine 8 years later - without piping. Hillbilly but works perfect.

If I were doing my house I would run it 7-8 inches off the ground.  Just because that is usually a spot that has little chance to be nailed/screwed.

Actually I wouldn't run cabling in my house.  You run cables now and in 2-3 years wireless is faster.  Seriously what could you be doing at your house that wireless wouldn't work?  They make 1GB wireless routers for under $100.  Soon 10GB will be the norm.  Your internet speed - if it is blazing blazing fast - goes at 100 MB.  That is 10 times less than your 1GB wireless throughput.  Also your computer will only be able to process so much data being saved to it.  I have my house teched out to the max (video/music hub server, print server, 10 devices online, virtual servers, mac/pc) and I haven't thought for a second running cables.  Unless you are looking at selling your house soon to a couple in their 40/50s (the only demographic where cabling was known to) I see this as being pointless from a pure tech standpoint.  My advice - take all the money that you were spending on this project - read reviews - buy a good router, buy a repeater if you are worrying about reception to devices, buy a central hub device (NAS) to store and share your stuff and enjoy.