Nikon D4, Nikon D5200, or Nikon D800?

Which is best for wildlife, action, and pet photography?

The D4 would be the best for action easily. Wildlife would be the D4 if the wildlife has a tendency to move (especially in random, skittish ways), or the D800 if it doesn't move too much or moves in predictable patterns.

Pet photography? If it's a tame pet which will sit around and let you take its picture, the D5200 will be more than enough.

D5200

Save your money for the lenses you will need.

The D7100 is the camera I would recommend and add these lenses to your wildlife kit

* 200-400 mm f/4
* 300 mm f/2.4

The D7100 will give you the extra reach you need when shooting wildlife.

How many times are you going to ask this question? Why don't you just read the answers you've already been given?