For football, I think you will really struggle, with the f4, once the sun goes down. I've shot a lot of night time high school football and shoot from the sidelines. Some fields are better lit than others, but none of them are lit well. I primarily use the 70-200 f2.8. Early in the season, when the days are longer, I've had some success using the 1.4 extender on the 70-200, but once the sun starts to set the extender comes off. Under the high school lights at f2.8, I'm usually shooting iso 3200 with a shutter between 250 and 320. On a number of occasions, I've tried using 135mm f2, but the lack of zoom means having to reposition too much for my taste.
I did have the opportunity to shoot a high school football game at a professional soccer stadium. Those were some of my best pictures. I took two cameras, one with the 70-200 f2.8 and one with an 18-200 kit lens. They both did well, but the pictures from the 70-200 were incredible.
I also shoot high school basketball. I find myself using the 24-70mm f2.8 and the 85mm 1.8 on my 60D. I get more keepers with the 24-70, due to the flexibility in zooming, but when one hits on the 85mm, watch out.
I think you could get away with the 70-200 f4 in the gym, but that extra stop is nice. I'm usually shooting iso 1600 with a shutter around 320. All else being equal, you would need to shoot iso 3200 with the f4.
I primarily use the 60D for sports due to the faster frame rates, but occasionally I take my 5D Mark II. On the 5D, for basketball, I prefer the 70-200 f2.8.
One other thing I should mention is that a lot high school football fields and basketball gyms are better lit in the middle, then they are on the ends. This is an issue, since the scoring happens at the ends. You should plan on a difference of one to two stops, between the middle and the ends.
Good luck with your decision!
Do you think the F4 IS will do good with night time football games? I have not really shot much above ISO 2000 and with the ancient lenses I have, they turn out pretty bad! I really loved the IQ of the F4 IS I rented for my trip, but will it blur the background enough when shooting senior photos? I normally borrow my brothers Tamron 17-50 F2.8 when I do those.