May arduino hang when current is changing over time?
There really isn't any such thing as "Current driftage". If you draw too much current from your battery it will cause the voltage to drop, and if it drops too much, the Arduino will stop working.

If you have high current devices like motors being switched on the same power supply as your Arduino you should add a reservoir capacitor as close to the input of your Arduino as possible. That will even out short term drops and spikes in supply voltage as your motors switch. 

If you draw too much current continuously then your supply voltage will drop and stay dropped, and a reservoir capacitor won't help. In that case you need a higher current battery (or separate power supplies for your Arduino and motors.)

Edit: From your question it's pretty clear that you don't understand voltage and current and how they relate. I suggest you read up on the subject.

In short, the amount of current that flows through a component is determined by the voltage divided by the resistance: i = v/r. (Where i = current in amps, v = voltage, and r = resistance in ohms.)

The current that will flow will be limited by the max current that the power supply can provide. 

For a non-regulated supply like a battery, the voltage from the supply will drop as you draw more and more current. 

For a regulated supply, the supply keeps the voltage constant up to it's maximum current limit.