Circuit for controlling 3 DC motors
Your system looks right, this is how I would do it.


If you had a motor driver smoke out then the voltage was too high or the current was too high or no back EMF protection. If you plan on running the motor at 20A max then your driver should be capable of dealing with 24A, at least. Don't forget that even if your motor draws 20A when stalled there is probebly is very short 30A-35A 'spike'. Always plan on 10%-30% more when selecting components.
Both option would work but it would be better to use a single battery for the sole reason that your system will last longer. As long as you have large enough capacitors on your driver there shouldn't be any problem running all your motors from the same battery.
Again, if you plan to run your motors at 20A it would be better to use a 14AWG wire that can deal ~30A. If this is a hobby project and your wires are not many meters long please don't save 2$ on wires. If you are going to make a million units then your should go for a 18AWG wire as this might be enough if you draw 20A for a very short period of time.