Mot. . .
To answer your questions:
"Does time exist outside of a gravity well?" Yes, time exists everywhere, even in the meanest nastiest black hole. It just moves more slowly FROM THE PERSPECTIVE OF ANY OBSERVER OUTSIDE THE BLACK HOLE. The view from the inside would be the opposite, the rest of the universe would appear to speed up.
"If you are in a space station that is spinning to give artificial gravity, then will spinning the space station always cause artificial gravity? " The centripetal (or centrifugal, depending on where you are in the system) force mimics gravity but it is not. It's just Newtonian necromancy. And yes, no matter where you are in the universe. . .even in an area of space so empty that galaxies are not visible. . .it will work exactly the same as in free fall or at a la Grange point.
"My guess is that in order to have the acceleration, the space craft has to be in a gravity well. " This statement makes no sense to me. If you mean the radial acceleration of the spinning body. . .no, it has nothing to do with a gravity well. Don't forget that if you entered your hypothetical spinning station at the axis and never touched the sides, you would be in free fall until you grabbed something moving and let it pull you around and around.
If you did it just right using suit thrusters, you would hover just above the moving 'floor' of the space station while people standing on the 'floor' would zip past you. This scenario assumes no air in the station, by the way. The moving air inside would eventually cause you to 'catch up' with the spin of the station and you would 'fall' to the 'floor.'
"The corollary question is, does a change in the gravity well affect the perceived gravity inside the space station?" The answer to this is probably 'no.' If you are in a gravity well and the spinning space station is in free fall--either because it's in a stable orbit or because it is freely falling toward whatever massive body is the source of the gravity. . .then there would be no effect at all. But if thrust is being applied to the spinning station to keep it from falling into the gravity well, that thrust would tend to tilt the forces inside the station in a direction opposite that of the thrust.
Does any of this make sense?