Charging board for 3 battery 3.6V used for supply an Arduino board
Note that you cannot just put lithium cells in parallel without balancing them first (to make sure they are equally charged).

Also, you cannot just charge Li-cells by applying a voltage, if you do that, don't be surprised if the cells get too hot or even explode ! You need a dedicated charging circuit or module.

But let me propose a much simpeler solution:
Just balance your cells, then connect them in parallel. Charge them using a charging module, feed that module form a 5 V power supply. These modules can easily be found on ebay.

To make the 12 V for the LED strips, use a DCDC boost converter. On ebay you can find these and most have a trimmer to set the voltage to the 12V you need for the LEDs.

You can also feed the arduino from this 12 V, make sure you connect the 12 V to the power adapter socket (Vraw) so that the voltage goed through a regulator which will supply 5 V or 3.3 V (depends on your Arduino module) to the microController.

You can make it extra fancy by monitoring the battery voltage through a voltage divider and feed that to the Arduino. When the battery voltage gets too low then the Arduino can detect that and switch off the LEDs so that you know its time to charge the batteries. :-)