Li Ion series Charging and Discharging
Fully charging LiIon batteries involves 4 phases of alterning constant current with constant voltage (see second link below). However, if you keep charging at "slow rates", it's easier:


Charge at constant current of 0.18C or less.
Stop when battery outputs 4.2V. It's charged around 70% to 80%.
When battery outputs 3V, it's considered discharged. Going below this value may harm it.
Your batteries hold 2000mAh, therefore C=2000mA, therefore the "slow rate" charging current should be 360mA or less.


See:


http://www.candlepowerforums.com/vb/showthread.php?294844-Charging-Lithium-Batteries%97rather-Balanced-and-in-Series-or-Parallel
http://www.powerstream.com/li.htm
http://www.instructables.com/id/Li-ion-battery-charging/


It is unwise to charge LiIon batteries in series if you only check the total output voltage (12.6V). Due to slight differences between batteries, even of the same made and manufacturer, some of the batteries will be charged more than others, and go beyond the safety mark of 4.2V.

Usually you use a balancer to prevent this problem. Depending on your goal (are you building an UPS of some sort?) you may prefer to have the batteries in parallel, and use a Boost converter to raise the output voltage.

To anwer your questions directly:


LiIon cells of 2000 mAh, even of C=1, are able to provide 2A continous. So you're within safe limits with your load.
A 'charger' can be of lots of types. If charger is LiIon prepared, then it looks right. If 'charger' is just a power source of 12.6V, the you can't use it. It will force way too much current into your battery pack, without caring about slow rate or charging phases.