I disagree with the guys who say a tank is going to slow down your filling up of your tires. A tank is required for air horns and air tools.
If you keep the compressor on and tank charged all the time you are on the trail or turn it on a few minutes before you stop to air up you will already have 150 PSI of air waiting to get a jump start on that first tire and any time you pull the chuck off to check the tire pressure you will get a couple extra pounds in the tank, same when you switch tires, to help that much more.
Now at some point the tank and tire pressure will be close enough that you will be relying only on the CFM of the compressor.
With an electric compressor the output is voltage dependent so I would keep the engine running if you want to get the absolute most out of the electrics.
IMO the biggest tank you can reasonably mount is best.
I have never been a fan of the CO2 set-ups...things can leak, they WILL run out and you have to make a special trip to top them off instead of just flipping a switch.
Not sure how long you can run air tools on one but we run air drills, grinders, wrenches and blow guns and fill air mattresses and might air down/up 6 times during a trip with air so we might likely kill a bottle of CO2 on the trail before we ever got to the end of the last day. They are like using a battery to run your fridge but not having a way to recharge the battery while away from civilization.
To each their own.
Darrell