Personally, I'd be inclined to go with the 70-200mm f/4L IS over the 70-200 f/2.8L (non-IS). The former delivers better IQ, and is a better deal (the f/2.8 is only $100 more new). I find that IS is pretty beneficial in a telephoto lens.

The 70-200mm f/2.8L IS II is an amazing lens, one of the sharpest zoom lenses avaialble. The only downsides are the size/weight and the cost. Note that there are five 70-200mm L-series zooms available - f/4 non-IS and IS, f/2.8 non-IS, IS, and IS II. In terms of IQ, I'd rank them f/2.8 IS II > f/4 IS > f/2.8 non-IS > f/4 nonIS > f/2.8 IS. If you absolutely need f/2.8 (indoor sports) and your budget doesn't cover the f/2.8 IS II, then the f/2.8 non-IS is a good choice (for indoor sports, you would generally want a high enough shutter speed to recude the need for IS). Otherwise, I think the f/4L IS is really a great lens.

The 70-300 L is new enough that you won't see too many of them on the used market.

Video quality of the M is better than a P&S - it'll be basically the same as your 60D, but the on-sensor phase AF allows AF during video.