The little metal bit on the end of the coaxial cable which screws onto the modem is know as a "PPC EX6 4904" connector. I'm not sure if there's a simple name for it. It can be unscrewed from the modem but you cannot physically remove it from the cable as it has been stated above that the connector is crimped. As far as I know you would actually have to cut the cable, feed it out through the wall and back in where ever you want it, then put another of the aformentioned connectors on and crimp it again.
I wouldn't advise inexperienced would-be electricians to do it though, and Telewest certainly don't condone it. If you know any electricians who have said experience, it may be worth asking them.
Alternatively you could go for the easier option and get a router.
And in answer to your first question, no the house electricity doesn't have to be off. Whilst the coaxial cable is disconnected from the modem, it has no power source.
That is of course, if you're talking about this connector:
No, you can't simply screw that off. The cable has to be cut and another connector has to be crimped onto the cable once it's in its new position.