Whew - long read!

Sports and birds/wildlife are probably the two most expensive subjects in terms of gear. For the body you want great AF, fast frame rate, and short shutter lag, and for the lens you want long, fast, and likely both.

So, you'll obviously need to make some compromise choices.

IMO, there really aren't huge, across-the-board differences between Canon and Nikon. You can consider DxOMark, but not much - they score the sensor and only the sensor, their normalization process has flaws, and their Overall Score is a black box calculation. Consider - if a sensor offers a slightly less noisy image at high ISO, but the camera the sensor is in misses focus or has a slower frame rate, which is better - a crisp shot of the key moment with a little more noise, or a shot with less noise that's blurry and a fraction of a second too late?

While I can't say one brand beats another overall, for specific use cases and budget levels, there are differences. For example, if your main interest was wide angle landscapes the Nikon D800 + 14-24/2.8 is great, topping Canon's corresponding offerings. For birding, at the extreme high end, there's no huge IQ difference (1D X vs. D4), but I'd give Canon the edge for their much lighter 500/600 II lenses and having an 800/5.6 which Nikon lacks (but the D4 can AF at f/8, meaning the 600/4 + 2x can AF).

But on to stuff you care about...

In the midrange, Nikon doesn't have anything that really matches the Canon 7D (great AF and fast fps). On the lens side, the Canon 100-400 is much better than the Nikon 80-400, Nikon has no 'affordable' 400mm prime to match Canon's 400/5.6 (a great birds-in-flight lens), and if you need sealing, the 70-300L has no real Nikon counterpart. IMO, the 7D + 100-400 is just about the best midrange birding combo, and can be had used for just over $2K (there are 7D's and 100-400s on my local Craigslist in the $1000-1100 range right now).