If the game needs to set anything up (like if something in DX 11 was removed in DX12), it should install the runtimes for the DX version it needs.
For example, I am on Windows 8.1 (which ships with DX 11) but I still play lots of DX 9 based games. When I install those games, they run their own included copy of the DX 9 runtime installer so that the game will run properly. In the case of Windows 10 and DX 12, the same should apply.
Won't improve performance at all... just because the OS ships with a newer DX version doesn't mean older games on older DX versions will suddenly use the newer version and run better because of it. Doesn't work that way.