Do my doors need to be good quality to fit the door handles?

If you have solid wood doors (44mm +) then screw fixings will be adequate for both levers on rose and backplate.

 

If you have thinner doors or hollow doors the back to back bolt fixings are essential with the levers on rose to give security to the handles. If you try to screw levers on rose door handles to these cheap doors, they can come off in a few months. The reason being that on thin doors you will need to use short screws which will not take a firm grip into composte doors.