Suppose the energy transferred to a dead battery during charging is W. The recharged battery is then used until fully discharged again. Is the total energy transferred out of the battery during use also W?

No

Explanation:

The amount of energy transferred out of battery will not be the same as given during charging.

A battery has its internal resistance as well which will draw some energy, through this internal resistance energy is lost

P = I² R

where P is energy, I is current passing and R is resistance.

As per law of conservation energy will be saved as it is the sum of drawn energy and energy wasted in internal resistance. But the total energy transferred out will not be same.