Matt, welcome to CNCZone. Your research and understanding of the interaction of all the components is correct. Of course there are many variations in components, etc, but you are correct with your general assumptions.
No, not all boards, controllers, steppers, servos, etc are compatible tho there is a lot of compatibility in like devices. IE, a lot of stepper motors will work with a lot of the controllers. It is pretty easy to find out if a particular stepper or servo will work with a particular controller. Easiest way, is do a search here on CNCZone or ask in one of the forums. Someone probably has tried your combination or knows for sure whether it will work or not.
Mach3 is a popular software tool for controlling the CNC machines. It is very flexible and very powerful. Support is great both from ArtSoft as well as the many, many users here on CNCZone. There are other software solutions like EMC2 that also work well. Mach3 is not free tho you can try it out for free on a limited basis (I think it has a limit on the number of lines of gcode it will execute) but you can get a good idea of how it works. Although it is not free, I think most will agree it is a heck of a bargain for the quality and functionality you get.
Hope this helps