Technically speaking a ship travelling at warp speed has no inertia as we understand it. That is why the warp engines must operate the entire time the ship is travelling. Otherwise the ships captain could accellerate to warp 9, shut off the engines, and simply coast to wherever he wished to go.
Essentially the ship is standing still, and space is moving around it.
The warp engines don't provide forward thrust as I understand it.
They merely create the subspace field that warps reality and enables the ship to travel faster than light. A seperate propulsion mechanism is still required to actually move anything. If the field goes down, you drop(rather rapidly) to your "real" top speed, which is a fraction of c.
So yes, inertial dampers keep the crew from becoming so much strawberry jelly when they jump to warp.
They also let them rapidly change velocities*, whether in warp or out of it, enabling at-will selection of speeds as well as many maneuvers that would otherwise tear the ship apart.
*Velocity includes speed AND direction, just in case someone didn't know.
Yes, unfortunately the exact definition of how the warp, impulse, and maneuvering thrusters work and interact depends on which episode of which series you are watching.
But in general the concept of a warp ship involves one that travels without velocity, either through a series of teleportation leaps, or by warping space (contracting the space in front of the ship and expanding the space behind the ship, which causes the space the ship itself is in to move (taking the ship with it). Neither one of those concepts has velocity.
But then there is also the happy simplistic "Warp drive is just like regular drive, but faster than light" idea, which would have momentum, but doesn't seem to be possible using any math I can comprehend.
Then there is the Star Wars (and others) hyperspace idea, which achieves is slower than light, but travels through an alternate dimension.