Design to whatever your output resolution is. For example if you are designing for 1080p you will want to design your files at 1920px × 1080px.
Video backgrounds are more than welcome! We prefer files in an H.264 format. We can also convert any video file you send us into the proper format. If you've designed the background in After Affects send that along as well so we can try to make sure we get a perfect loop out of it.