Controlling 12 Solenoids from Arduino / ShiftOut

FET: IRF540N

No.
Not only is that vastly overpriced, it is not a logic level FET, it takes 10V to fully turn it on. You want a FET that takes only 5V. The gate threshold voltage is where it just starts to turn on not when it is fully on. For that see the gate voltage in the Ron or drain source resistance line of the spec.