They definitely do. If not, you wouldn't see anyone using Vista for a gaming system. You've always been able to play previous version DX games on the latest DX version.
Yes, it will. Although there is a little quirk where you may need to install the DirectX9.0c components. It doesn't overwrite the DX10 ones, just adds to them.
Yes, Vista actually has both DX10 and DX9 subsystems, as DX10 is not backwards-compatible like every other version - so you have to keep both up-to-date.