I think the hypothetical answer to the hypothetical Warp Drive is that Captain Kirk would not experience Time Dilation because that is a function of velocity in "normal space" where as when Scotty has us chumming along at Warp Factor 6, we are "outside" of "normal space" and therefore time dilation has no effect on us. That miraculous "sub-space" permits us to go faster than light, AND do it in "real time".
Faster-than-light travel causes worse problems than just time dilation; it could cause time paradoxes and reversal of causality. Some people say that it always causes such problems, although I think that you could sometimes be able to use a FTL drive without causing a reversal of causality (assuming , of course that you could use FTL drives at all, which is looking very unlikely).
Ships always travel at the speed of plot -- even in the more physics-oriented Babylon 5. JMS was fond of describing a ship's speed as the speed of plot, although he sold himself a little short; there were actual efforts to work out a realistic transit time in many cases (although the plot was always more important, which it kinda has to be; the single most important thing in telling a story is the story itself, and if you let yourself get tied up in details, the story will suffer, so although its best to have physics and plot coincide, when this cannot happen the plot has to win).