Whether a battery stores energy or simply it converts chemical energy into electrical?
A battery is a chemical reaction. A simple cell will produce charge by ion exchange and consume electrode material. A depolarizing chemical in the cell keeps this reaction going. The rate of the chemical reaction is increased by drawing current from it -  you alter the equilibrium of the reaction. If you draw current or not the chemical reaction continues and so the cell will have a limited shelf life. Usually the reaction is in one direction only but in rechargeable cells the reaction can be reversed when current is fed into the cell. It doesn't store the charge as such but uses it to reset the chemistry.

Fuel cells consume chemicals (typically hydrogen and oxygen) to produce electricity.