It's got a COIL - which means it's NOT a Piezo.

What you have is basically a poor-mans Loudspeaker (as opposed to a Buzzer). It's frequency response is dependent on it's mechanical construction.

OHMS LAW... how much current is it going to suck?

Is it going to generate a Back-EMF?

Your poor Regulator at the higher voltage supply is going to have a hard time feeding the PIC and the Sounder which incidentally will most probably introduce a lot of horrible noise on the Supply line adversely affecting the PIC.

This is a poor choice of device for any low-current application. For best efficiency, you're going to have to adjust your circuit and drive the device more like the way you would an Audio Loudspeaker.