Canon EF 100-400mm f/4.5-5.6L IS II USM Lens or other options?

I had the 5dIII....when I got my 1Dx, I really never picked it up again.  Then I got a 1DxII and then a 5DIV.....and because of the 5DIV I really havent picked up the 1DxII since.  The newer cameras with better ISO performance really help.  So there is my rant...now to answer your questions...

1. The 70-200 IS II is a real GEM.  Without knowing where you stand and take the photos its hard for anyone to say that 280mm is going to get you enough reach.  But given its price advantage for you, its worth the try assuming you can afford to be out the money if it doesnt have the reach you want.

Having a camera with higher MP would help because you could crop for what you want and not lose the detail or begin to pixelate.

2. To me, unless you can afford to run at higher ISO because the cameras capability is there, you could go with the Sigma lens, but I wouldnt because you will always want more and more lens speed, either to compensate for lower light or because you want faster shutter speed to freeze the action.  The kids are only going to get faster....

3. quick answer No.  If you snap a photo realize that the amount of light coming in is going to hit the exposure sensor and calculate what the lens opening should be based on shutter speed/ISO (I am assuming that you would set the shutter speed via Tv to make sure that you have the correct shutter speed and not something that the camera picked.  I would probably set the camera in manual and pick shutter speed and opening, letting the camera pick the ISO.  That way I am making sure I am getting the key settings I would want (narrow DOF versus a deeper DOF, panning the camera to get the sense of speed versus a frozen snapshot).  At the end of the day you want to have a properly exposed photo.

4. First off before I spend any money, I would see if I could borrow or worse case RENT the 1.4xIII and try it first.  This way you know if it gets you what you want.  If not then you know you need to try a 100-400 and see if it gets you what you want.  I am a Canon fan, I only have Canon lens so it is hard for me to think through saying you should get the Sigma lens.....I just dont think it will get you what you really want.  What ever you decide you need, I would suggest buying a used one, people are always moving in and out of gear and you should be able to find a good one.  Just remember that you might have to MFA the combo of multiplier and lens where you didnt have to before.

I hope this helps...I was a little fragmented in my thoughts so hopefully it makes sense.  There is so much that I take for granted because I have been in photography for almost 50 years that often many things come as a second nature to me so if I need to clarify something, please ask.  No question is a dumb question.....