Arduino board is a active element, ohm's law apply but little bit tricky in calculating power consumption. For Arduino board case, using linear regulator, it only draws current it needed (around 40mA without external shield connected) for a range of input voltage.
For converting 12V input to 5V VDD, power distribution is:
A) Total input power 12V x 40mA = 480mW
B) Dissipation on voltage regulator (12V-5V) x 40mA = 280mW
C) Dissipation on ATmega MCU and peripheral components 5V x 40mA = 200mW
If you add a led shield, current consumption goes high (>40mA) and power dissipation in both B and C part will increase proportionally. But, if input voltage increase (>12), only power dissipation on voltage regulator B increase.
Let's go back to your questions:
- An electronic device, when it's connected to a battery, only gets the amp required or the battery gives more and the device breaks?
Ans: It only gets the amp it required, for linearly regulated circuitry.
- A voltage stabilizer is good or i need one with current limit?
Ans: For the case of car battery, voltage pulse can up to 14V with running engine, stabilizer is recommended. From my experience, some voltage regulators cannot sustain such a high voltage drop, let's say from 13V to 5V. And high power dissipation across it (as in B above) will create heat and burn the regulator ultimately.