I think "invest" is the correct word. In the context of the financial industry, "invest" usually implies that you have purchased actual stock in a company. But in general, the word means that you have committed something of value to someone else, and hence, your well-being is tied to theirs. We also say that a teacher is invested in their students success because they have committed their time and effort to help those people succeed. In this circumstance, NASA has given SpaceX money, and if SpaceX does not succeed, that will hurt NASA. If SpaceX does succeed, that will help NASA. NASA could have, instead, used that $381 million to try to do it themselves; that money is now gone, and if SpaceX fails, NASA won't get a do-over. For those reasons, I think it's accurate to say to then say that NASA is invested in SpaceX.
Again, I disagree. I think most people will understand it in the way I said above, because for most people, "invest" does not imply "bought shares of." Because of your involvement in the startup community, you tend to assume a specialized definition of "invest" rather than the general definition that most people assume.
Which is related to what I said about being plugged into the startup community. This article, however, appeared in Wired, which is a more general magazine.
Right. I shouldn't have continued the implication, but in a generic sense their stake is still valued at $380 (with perhaps some recoverable if Elon Musk suddenly disappears and SpaceX creates the Blob or something). I guess I was distracted thinking about the upcoming IPO. Last I heard was 2013Q1.