With any game the first thing to look at is the "bare minimum" requirements. Once those are met you shouldn't see problems unless you are trying to run it above what your current hardwares support. The screen there is from one game that includes a DX9 installer since that is a minimum requirement.
You can install a newer DX on a system with an older card. But you can't run an older Direct X version then what a card requires. Besides DX you also have to look at the other things found in newer games that are not supported with the older cards. Anisotropics, antiliasing, pixel shaders, etc..