Hi,
I guess 2A is wrong at all. It should be 2Ah. It's not capacity, it's not current, it's charge.
500mA is the current.
There are batteries with 2Ah, which can easily supply 500mA, and there are 2Ah batteries where you are not allowed to draw 500mA.
As so often: the datasheet should tell.
Connecting a load to a battery will always drop the voltage. In short time (seconds) according Ohm's law (battery internal resistance and wiring resistance) ... long time (over hours) according the charge state.
All this is well known and you can find a lot of information in the internet. High quality information is provided by the battery manufacturers.
(They know best how their batteries work)
Klaus