Can I use different sized steppers for different axes? So basically, I have 3 different types of steppers.

A NEMA 23 stepper for the Z-axis,
a good quality NEMA 17 for Y-axis and
another lower quality NEMA 17 for X-axis.

This setup should work right?. I'm using TMC2209's stepper drivers and all are well within the drivers rated phase current limit.
Each axis is fine using a different stepper size and/or quality. You will still need to tune their operating current and steps/mm for each of course. Make sure your motor mounts fit too.
Where it might become challenging is if you wanted two different steppers on the SAME axis. As in, a NEMA17 + NEMA23 for a dual Z axis. I don't think this is what you intend though.