Also, according to Star Trek TNG, the warp speed is not linear.
Using the TNG calc it would take 78 days to travel 1400 light years.
EDIT: ^ at warp 9.9
TNG had warp 10 as a sort of "speed of light" though, whereas TOS could go over warp 11!!!!!!
The only way (according to current physics) to travel faster than the speed of light would be to physically warp space (aka warp drive) ... this isn't really travelling faster than light but instead manipulating the universe to make the distance traveled shorter. See also: wormholes.
No, not at all. "Warp drives" move the space-time around the ship itself, the "warp bubble" would indeed move faster than light*, the vessel would just remainn stationary within that area of space-time. Wormholes would actually decrease he distance, but through a totally different mechanism.
*There's no known speed limit for space-time, hence expansion.