It would not be good to use 4 x AA batteries (regardless of the battery chemistry) with the D24V22F5 regulator, since as those batteries drain, their total voltage will drop below the dropout voltage of that regulator. You can learn more about why that is important under the "Typical dropout voltage" section of that regulator's product page.
As for your second question, a typical 9V alkaline battery is impractical for motor applications, so we do not recommend using that as a motor supply. You can learn more about why in this blog post about understanding battery capacity.
I followed your link and it looks like that pump draws 3A when supplied with 12V. So, you should probably use a regulator that can comfortably provide at least 3A continuously. I cannot tell if you are committed to providing 9V to the pump, but if you let me know what you have decided on, I might be able to offer suggestions on an appropriate regulator and battery.