choice of stepper motors

I always find 42 stepper motor for x and y axis,
I am wondering if this is necessary to have such large motor for lightweight motion such as x and y axis,

If there is any choice for smaller stepper motor?
Does smaller mean cheaper as well?
Does it have to be compatible to motor driver as well, as far as I know most mainboard are using A4988 or DRV8825?

Thanks

I recommend to use the same motors for x, y and z axis.
You can look for them on ebay. There you will find offers with a bundle of 4 or 5 stepper motors relatively cheap.
The only thing important for the stepper drivers is that they can handle enought current (most stepper motors for 3d printers are under 1.5A and can be used bipolar. They can be used with the standard motor drivers)

Motor sizes are determined by torque required to move their load and the way you want to move it. For a given load, higher acceleration means higher torque is required.

You can figure out approximately how much torque you need for your printer by using : Motor Sizing Tools It can be a little intimidating at first, but play with it a little and you'll quickly see it isn't so hard to use. Don't assume that all three axes have the same requirements. Calculate what is required for each axis. Once you know the required torques, shop for motors that provide those torques.