What are you hoping to record? The 70-200mm focal length (which I've only used briefly on the 5dII) is wickedly telephoto for general purpose video. The image is amazing, at least on the f2.8 II and presumably on the f4 as well. For like wildlife or whatnot it would be wonderful and it's nice for portraiture and music video inserts, possibly, but those aren't the most useful focal lengths for most video.
Not to mention a telephoto lens increases perceived skew. So, if you don't have a tripod...yes, you would want IS very much. But really you want a tripod. With a real fluid head. Everyone skimps on this. Don't.
For sports a long lens is great if you're by the sidelines, but if you can get close to the action longer lenses are way more exciting. For narrative stuff, 24-70mm (on the 5dII or 17-55mm (on the 7d) is almost objectively the best choice. But for fun, dynamic stuff, the wider the better...
The sharpness of an L lens really won't matter that much for video, but the lack of distortion, the magic color (which isn't real), the wide f-stops, the lack of fall off, the good ergonomics, and the good bokeh will. But any decent lens should be fine at regular stops. Even the lowly kit lens and 55-250mm IS do great for video (even at f5.6 excepting the 55-250mm's insane fall off, but no worse than the overrated 17-55mm IS) but they're quite slow.
Depends what type of videos you'll be shooting.
My recommendations are:
simply because I film snowboarding/longboarding and I need a good regular telephoto as well as a nice wide lens.
IS helps in certain instances, but really it's not too noticeable in my opinion and I don't always use it.
Not bad advice... I'd throw in the 24-105mm f4 L as a general purpose lens for the 5dII if you get that, especially since you won't need the extra stop so much outdoors.
Seriously, though, buy based on focal length and not "L" vs not. What focal lengths do you use most now? Do you want to go wider or tighter than your current lenses let you go?