So I haven't looked at the various Slingboxes in a while but it looks like you can control your cable box with the Solo (any of the current Slingboxes will do that as well as the discontinued Classic) and that includes controlling your DVR.

Think of the "Classic" as Slingbox SOLO but with a coaxial cable input and without the component video/HD inputs/support. And instead of having separate RCA A/V jacks like the Slingbox SOLO, Classic uses a combined A/V port that you split into the separate jacks with the included cable (so make sure if you buy the Classic you get those cables, there's two of them). Best way to make the comparison is just to simply google it and see for yourself. The slingmedia website also does a good job of describing the differences between their various current boxes.

Finally, none of the boxes can stream HD but the SOLO and PRO allow you to input (and pass through) HD sources and that does result in better streaming video quality.