Can 12v 3-12A battery power safely mega 2560?

Yes but....

First, the current capacity of the source (battery) is irrelevant unless it is too small. Your AC line into your home can supply 15Amps to any given plug. If you plug a 7watt light into the socket the light only draws a small part of an amp.

The input voltage range of your board is 7V to 12V. 12V is at the upper limit and could cause over heating (and potentially failure) of your board if you power anything of significance from the boards 5V output.

I recommend you follow dozie's advice and get a power capable device to lower the voltage to 5V.