If you have a cat in a cylinder of air in space, and push it toward one end of the cylinder, I think it would get those signals since the cat is being pushed through the air.
Having done "zero-g" in aircraft many times, you do feel it. Your blood shifts. Your spine decompresses. Your head gets lighter. You very much know that you are falling.
A lot of confusion here probably comes from the fact that g-force is not the same as acceleration. It's apparent acceleration. So 0g in an aircraft is entirely possible. So is 0 acceleration but that's not 0 g, that's 1 g.
the cat experiences "zero-g" as it accelerates towards terminal velocity
I think you meant the cat experiences 'zero-g' AFTER it's done accelerating and has reached terminal velocity.
I'm guessing in most cat falling situations the cat does not reach terminal velocity, so it is accelerating the whole time and can adjust based on the direction of acceleration.
No. Terminal velocity is one-g, the same as sitting in a chair. A skydiver at terminal velocity feels exactly as much force/acceleration from the wind as they would feel force from lying down on the ground.