I know a few people using integrated graphics just fine with Windows Media Centre. I'm running an Atom based nettop with only 2GB (!) and it'll handle 1080p fine as long as it isn't streamed.
Personally I would get the CPU with the Intel HD4000 which is more than adequate for your needs. Something like the Core i3-3225 would be great and not too expensive.
Don't forget that some of your memory will be required by the GPU, so grab 8GB if you can and allocate 2GB to the GPU. To be honest, 4GB (with 3GB for Windows) should be fine too - but the difference in price between 4 and 8GB these days is minimal.