Arduino powering an external circuit with higher voltage
You are seeing this in wrong way,,
1)if a device is rated at 12V (say a relay) you can use it with various voltages defined by the manufacturer if you refer the "Datasheet" of the product it will tell you what is the minimum voltage (say 9V),Typical voltage(say 12V) and Maximum voltage(say 14V) that the device should be operated.

2)So you even if u connect a 12V device to a 5V battery with 10A it simply wont power up

3)Ohm says V=IR is only applicable to simple linear circuits if you observe this carefully there is an R so if u supply a device lower voltage it necessarily wont drain less current as it is purely decided by 'R'. Higher the R lesser the current it draws ,Less the R more current it draws,,,which is in General cases.

4)You cannot get higher voltage from Arduino DIOP until u use a BuckUp converter either as the loss of current or bandwidth,there is always a trade off

Hope this helps...