Regarding your energy conversion question, it is entirely dependent on the circuit. Say for example you have your battery and have no wires to complete the circuit. A circuit that isn't closed is referred to as an open circuit. A circuit that has infinite resistance. In this case, your potential energy remains as potential energy.
Now say for example you have a closed circuit that creates an electric current. Depending on your circuit, most of your energy is converted from the batteries stored energy (potential energy) to heat (thermal energy). However, the rate at which this energy is converted is dependent on the circuit.
Let says you have a closed circuit that has ZERO resistance. This is known as a short circuit. In a short circuit, you theoretically have infinite power consumption, and remember, power is simply the rate at which energy is consumed or produced. So for an application that has infinite power consumption (or an infinite rate of energy consumption), you have the batteries potential energy instantly being converted to thermal energy. Things get really hot! If you ever short circuit a small battery with a piece of tin foil, you will notice that the piece of tin foil gets very warm. This is a perfect example of electric potential energy is converted to thermal energy.
Someone correct me if I'm wrong, but you could also say that the potential energy is being converted to kinetic energy due to the mass and velocity of electrons. But when dealing with circuits, the main concern is typically the thermal energy produced.
Regarding your second question of AC power, could you be more specific what you are asking?