Although I am making an electrochemical battery (1 lead acid cell), I thought that this question would be more suitable here. Would there be any difference in the voltage and current of the cell (when connected to a lightbulb) between a cell that was charged for 5 minutes vs. a cell that was charged for 10 minutes? I'm guessing that the voltage supplied and current will remain the same but only the time until the bulb runs out will be longer with a 10 minute charged cell?
The voltage of a cell is not constant and depends on the charge state. In addition, the ESR (the output resistance) of the cell also depends on the charge state, and when dealing with certain battery technologies, is even non-monotonic (This can actually be seen in very high current loads drawing from lithium-polymer batteries - the ESR will decrease first, resulting in a potential higher current draw and output voltage, even though the battery is less charged).