how easy would it be to use it to, say, provide low current power, or speaker signal or as you say TV coax?
You can supply up to 15W via power over ethernet (PoE), which uses 4 of the 8 wires for power, very commonly used for IP phones, CCTV cameras, wireless access points and the like. There's a new standard ratified to deliver 25W.
TV – something like this: http://www.cyberselect.co.uk/product/438 – although with a blank canvas it's much easier to get a proper aerial set up with regular coax going to each room.
Speaker signal – well it's copper wires so should work but would need to look at what the cable can handle. Have seen people comparing cat5 against audiophile stuff but they usually use 4-6 cables together, per channel. Things like the squeezebox devices are another approach to distributing audio around the house, and just work over regular cat5.