There is a lot going on here that needs to be considered. We'll take the issues one at a time.
I was thinking of buying the Canon 7D mark II...
The 7D mark II is a camera built around the idea of being a budget option for shooting sports. It does this very well. I've been shooting sports with the 7D Mark II for about 18 months now and it is by far the best APS-C camera I've ever used to shoot sports. It has a top notch AF system that can only be bested by an AF system with the wider baseline available to full frame cameras with larger mirrors.
The anti-flicker capability of the 7D mark II makes it highly capable under flickering lights as are found in many gyms and outdoor stadiums. Not only does it give you better shot-to-shot consistency in terms of exposure and color, but it times the shutter release with the peak of the oscillating lights. This allows you to use anywhere from 1/2 to a full stop faster shutter time for the same ISO and f-number than would be the case if you were centering your exposure in the middle of the lights' variation. For an in-depth look at the anti-flicker feature of the 7D Mark II, please see the case study at the end of this answer to a different question.
I was thinking of buying... the Canon EF 200mm f2.8 USM... is this a good (lens when used with a 16.X crop body camera) for various sports photography?
The EF 200mm f/2.8 L II is very nice at what it does. For the indoor sports, though, you're going to be awfully constrained by the 200mm focal length, especially on a 1.6X crop body. The complimentary lens with a shorter focal length is the EF 135mm f/2, but even that is almost too long with a crop body. By the time you've bought both of those you've already spent more than what an EF 70-200mm f/2.8 L will cost and you're well on your way to the cost of an EF 70-200mm f/2.8 L IS II which, in addition to the IS, is much better optically than the original non-IS design from the mid-1990s.
One option you might consider is keeping the 700D and the EF 17-55mm f/2.8 IS to use in combination with the longer lens. But the gap between 50mm and 200mm is still very wide. A 70-200mm f/2.8 is the most indispensable lens in the arsenal for indoor sports and even outdoor sports under the lights at night.
