First off be prepared to wait at least 14-16 weeks to get your permit. My app was recieved April 23rd and I got my permit last week on Aug 18th.
As far as carrying it in your vehicle I believe the gun can not be loaded and must not be in reach from the drivers seat if you do not have a permit. I believe the weapon has to be unloaded and in a enclosed carrier thus requiring two steps to fire it, removing it from the case and loading it. I could be wrong, but here in the state of Florida I wouldn't take any chances.
Kentucky permits are honored in Florida.