An electronic device, when it's connected to a battery, only gets the amp required or the battery gives more and the device breaks?
It depends on the device you are powering. If the device is designed to be powered with 12V then that is correct. However some devices are dependent on being driven at the correct current not voltage. For example an LED needs to be driven at a fixed current not a fixed voltage.
A voltage stabilizer is good or i need one with current limit?
A voltage stabiliser is needed if the voltage you have is too large for the device you have. A stabiliser with a current limit is only useful for protection if something goes wrong. Once the protection kicks in it no longer has the same voltage output.