I wonder what kind of training they'll be required to go through. A majority of NASA astronauts have had military training. How can you guarantee that an eccentric billionaire won't lose his cool once he's orbiting a heavenly body?

That's really stretching the definition of private company considering NASA is the major provider of both funding and infrastructure.

It's not stretching the definition at all.

You can hardly be a large business in any developed nation without having substantial business relationships with various world governments.

See: Microsoft, Cisco, Oracle, Boeing, Caterpillar, Lockheed Martin, Ford, GM, 3M, Berkshire Hathaway, General Electric, etc.

Might as well pretend Medtronic, Pfizer or Johnson & Johnson aren't private, because they do so much business with government healthcare systems.

I for one consider such companies to be quasi-government organizations. Especially to the degree that their products are purchased by the government via something other than a open market bid.

Dell doesn't purchase from Microsoft at an open market bid, either. No big organization purchases at open market bid if they can help it. In contrast, the U.S. government is frequently required to purchase on an open market bid (more or less, I'm not sure what your definition of "open market bid" is).

A lot of innovation was initially funded by the government especially in Silicon Valley. That said, Spacex is still legally a private company.

Many a time, when companies engage in capital intensive business endeavours they need someone to provide some incentive or even build the initial technology. NASA does this here much like DARPA did with the Internet's initial infrastructure.