My motherboard runs on DirectX 12, but my GPU is DirectX 11, and my OS is Windows 8.1 Pro (DirectX 11) so, will all these be compatible? Please explain how, if so.
Your motherboard supports Direct X 12 it doesn't "run on" Direct X 12. Its all compatible, and in fact your board was likely developed when Dx11 was the only thing around, its just selling "support" for Dx12 (which doesn't really matter its the GPU support thats important).
your motherboard? I think it depends on the gpu and not the motherboard.
anyway, directx is backward compatible. if you have an older gpu then it will not take advantage of the new features but it will work.
not sure if there are games that won't run if you don't have a dx12 capable gpu, but based on experience they will work just without the new features.
Your motherboard supports Direct X 12 it doesn't "run on" Direct X 12. Its all compatible, and in fact your board was likely developed when Dx11 was the only thing around, its just selling "support" for Dx12 (which doesn't really matter its the GPU support thats important).