Li-Ion battery remaining charge question
2000mAh does not mean 2 Amps.  It means if you draw 2 Amps you can do it for 1 hour before the battery is empty.  In reality, the slower you draw current the better the results.  That is, you might get 1 hour at 2 Amps.  But it is more likely you will get 2 hours at 1 Amp.  Or 4 hours at 500mA.

The maximum current a battery can produce has more to do with its internal resistance.  In an ideal battery there is no internal resistance.  There is no such battery in reality.  But if there were it would produce an infinite amount of current for an infinitely short amount of time.