Does the charge of a battery affect its discharge voltage or current?
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).