Which of these three lenses is best for an African safari?

I'm looking at the NIKON 70-200MM F/2.8 with a teleconverter, the NIKON 80-400MM F/4.5-5.6, and the NIKON 28-300MM AF-S F/3.5-5.6.
I want a very sharp, relatively fast lens that has a long range… Are there teleconverters available for the second two lenses?

Yes, the same teleconverter works for all three lenses.

You know, of course, that the fastest lens you've selected is the f/2.8. And that a 1.4x teleconverter will lose you one stop, and 2x will lose you two stops.

There are problems with teleconverters. They reduce the effective light-gathering capacity of the lens (the lowest f number goes up, the fast lens is no longer fast). More seriously with zoom lenses, at the wide-angle end the glass of the lens may crunch with the glass of the converter. I would not go that way. Consider a 500 mm mirror lens instead.

The 80-400. Unless you only want to take one lens in which case the 28-300.
A mirror 500mm mirror lens was suggested, now I use a Tamron 500mm f8, it's manual focus and a real PITA for quick accurate focusing. What would not be is the Sony/Minolta 500mm f8 which is the only autofocus mirror lens of its type - but you'd need a Sony body.
If you don't already have one, add a monopod with a ball head to your camera kit.

80-400 for sure. Super telephoto lenses generally are not as sharp as smaller lenses, thats just a characteristic of them. But those are pretty good, solid lenses with the 70-200 probably being the best of the three. However, the 80-400 will be your best option because of the versatility.