Yes, an object can have both of these at the same time.
Potential energy is energy that is stored in an object. Kinetic energy is the energy that is associated with motion. So what you have to have is an object that is in motion but still has more energy that it has yet to convert into kinetic energy.
A perfect example of this would be an object that is falling but has not yet hit the ground.
This object would have kinetic energy because it is in motion. But it would also have potential energy because it has a ways yet to fall before it lands.
Yes. Take a roller coaster for example - as the cart rides the rail it has kinetic energy. As it is moving it reaches the top so then it has both potential energy and kinetic energy.