Oh you didnt just ask me that ! Sweet ! Here goes:
The warp core is actually called the gravimetrical field displacement manifold and is used to create a bubble around the ship that would not react with normal space time hence bypassing the time-dilation issue with faster then (or close to) light travel. The Warp nacelles are the variable geometry warp coils or generators, so the official title for the warp nacelles is (and here goes) :
Variable geometry - gravimetrical field displacement - coil or generators.
To get a clear picture of what they do, take this analogy. ever seen those high divers at the olympics? triple backflips with double screws when they jump of a 10 meter high ledge? Gotta give them respect, I made a bad dive (2 meters) once and landed flat on the water. Man that hurt! Because the water is hard like concrete when I fell through because of the water tension. In the olympics you can see multiple little fountains splattering on the water just a couple of feet from where the divers actually hit the water. Why is that I always wondered until I found the answer. The breaking of the water surface tension by the water fountains made the smack the diver makes on the water MUCH less hard and actually could make the difference between life or death if the dive went bad.
So the Warp field does the same it actually softens up the space time continuum around it exposing subspace (like the fountains break the water tension for the diver and opens up the path to the deeper water) making the warp bubble with the vessel flow through space-time noticing almost no "friction". Untill you come upto warp 9.9999 with warp 10 being the end of the warp scale. (the borg had transwarp tech which makes a wormholes through subspace instead of using a lubricating warp bubble to flow through it)
I also have not found any other ship that uses the movable warp nacelle configuration. Maybe the prometheus class but i have not seen that in action, only on pictures.
To be homest i could think of the warp bubble being better aligned with the ship at warp speed. The warp nacelles are actually located under the saucer at impulse and in the middle of the ship when raised. But I have to agree it probably was the designers trying to find a way to differentiate from all other starfleet vessels and ... IT JUST LOOKS JUMMY !
Hope this satisfies your curiosity and the challenge is met ?
Thansk for making me think and do research (I love doing research)
Best
ASPHIAX
One final note : Miguel Alcubierre submitted a paper on real life warp drive !
It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the ``warp drive'' of science fiction. However, just as it happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here.