Its going to run from a NAS box to a Network Media Tank, so will be carrying data I guess?
Probably ethernet then, (which was my reason for asking,) in which case it doesn't matter to the cabling that it's 1080p video you're carrying encapsulated within ethernet, (ethernet and the cabling won't know or care what the content is, to them it's all just "data.") . In terms whether your infrastructure can handle it, it's then down to which flavour (data rate) of ethernet you choose. 100mbps (AKA "Fast" ethenet) should be OK unless you LAN is really busy, 1000mbps (gigabit) should be fine, but 10mbps probably won't cut (but you'll be hard pushed to buy 10mbps these days.)
The data rate of your video stream is more relevant than the fact that it's "1080p." For example, a Blu ray RIP might needs 40-50mbps of throughput to stream without breaking up, whereas an Internet flash video (which could still be 1080p) might be compressed to a "low" bitrate such at 1-2mbps (I've seen a demo of such.)
Also I have read that for this use, it should be crimped and straight into the media player, not terminated in a box and then patched cable to player. Is there a technical reason for this?
Nah, doesn't make any differences. Sounds a but urban myth'y to me. It's much "neater" to terminate permanent cable runs onto sockets and/or patch panels, then run patch leads from sockets to the equipment. If you've got poor terminations and/or dirty contacts it's more likely to prevent the link coming up at all rather than make it run more slowly.
There are arguments for minimising the number "connectors" in the pathway between devices (each plug/socket is an opportunity for dirt, mechanical movement etc.) but again, if you had problems with (say) a dirty contact, it's more likely that the link won't come up at all rather than work slowly at (say) 90% capacity. With digital networks, if you did get issues with a dodgy connection, the impact tends to be pretty dramatic rather than a gradual deterioration.
I just wouldn't get paranoid about it: Do a good job of installing and terminating you cables, avoid super-cheap materials and I'm sure you'll be fine.