Electronic – Replace 12V 2A centrifugal pump with 12V 1.4A peristaltic pump

I think that it's pretty safe to say that the 250V driving voltage used in the piezo pump won't affect your circuit. It's likely that the driving voltage is already isolated from the input, and even if you had a diode between the output of your FET, it'd have to be rated for > 250V in order to prevent it from breaking down. If the 250V coming back were a real concern, however, you'd want some kind of fuse that would trip at your MOSFET's rating of 60V so that it's physically isolated from any high voltages.

Lastly, the flyback diode isn't needed here. You're simply driving the pump with a PWM/analog signal. Assuming this thing is well designed, it would have any necessary protection already inside the box. I'd imagine some decoupling caps with the supply lines would be very beneficial, however.

The nameplate say the pump can be run from 6 to 24 V. Since 24 V is within this spec, yes, what you want to do should work.

Yes, the pump will run faster on 24 V than on 12 V. It will draw more power, but also pump the water faster. However, the overall energy spent to pump the same amount of water will still be higher at 24 V than at 12 V, even though the 24 V is on for a shorter time.