You probably won't be needing the more expensive, higher model Asus or MSi boards, as even the Asus Pro Gaming is very good. At the same time, the Asrock boards have gotten very favorable reviews, so I would guess that is also a fine choice. The K4 is actually a little smaller than a normal ATX board, so I don't know how it is to mount.
You will be able to play some games on the integrated GPU, though usually not the newest games. Turning down the settings to medium and resolution to 720p should allow you to play a lot of 3 or 4 year old games at good frame rates. I will of course recommend you buy a GTX 1070 or 1080, though if you're holding off until early next year NVIDIA will probably announce the 1080ti in January and I'd expect the 1080 to drop in price.