Most of Dish's programming is in 720P and 1080I, which is HD. 1080P is only available on Pay Per View. Bottom line, Component or DVI is all you need for 99% or your viewing. All this talk about the new 4K format and 1080P still ain't the norm. A HDMI equipped TV is required for 1080P.
Your Hopper will need a HDMI to DVI Cable for the picture. Or you can use the Component Out for the picture. Either way, you still have to use the Red and White Outputs for sound.
As far as your X Box, I think they use their own in house cable. Am I correct?
I have a old Sony 34" Wide Screen CRT,720P/1081I used with a Joey with a HDMI to DVI cable. The picture on this 208 pound beast is better than my new LCD's.