When you are using Impulse engines you are still under interial compensation so that the crew doesn't feel most of it movement. Personally I don't think that they should feel any of it but that doesn't look good on TV.
Full Impulse goes up to .25C and uses a low level warp field to reduce the mass of the ship so that a small impulse engine is enough. That way they don't have to stack the back of the ship with thruster bells like Star Wars.
Once you are at Warp you are not moving relative to the space inside the warp field but the space is moving relative to the space outside the warp field. Think of putting a ship model on a place mat and moving the mat instead of the ship.
Most of the time it is safer to generate a warp field away from a gravity well since that can cause wormholes and damage to your engines. That is the reason that they use impulse engines to pull away from a planet's gravity well before going to warp. If they were already away from one then they will just use their thrusters to turn and then go to warp.
A ship at warp has no acceleration. Some things like to say that if the inertial dampeners failed it would kill everyone. This is not true. It would mean that impulse travel would have to accelerate slowly but you could go in and out of warp without it.