Well, I know NASA is a gov't agency because if you're, oh, a private pilot and you happen to, well, let's say "bump" another plane in midair, you're required by law to report it to a federal agency. Those who aren't in the know will report it as an accident with the FAA. Those who *are* in the know will file an "incident report" with NASA. Because NASA doesn't investigate and the FAA does. The paperwork still counts, though.
As far as I know, the shuttle's like any other thing made by a private contractor for a gov't agency, like the Lockheed jets that go to the branches of the military, or the Pratt & Whitney engines that go inside the jets (and the rockets and whatever else).
As for point #4, here's the pertinent bit from the National Aeronautics and Space Act of 1958 (NASA's charter as a civilian agency "sponsored by" the government):
Sec. 202. (a) There is hereby established the National Aeronautics and Space Administration (hereinafter called the "Administration"). The Administration shall be headed by an Administrator, who shall be appointed from civilian life by the President by and with the advice and consent of the Senate. Under the supervision and direction of the President, the Administrator shall be responsible for the exercise of all powers and the discharge of all duties of the Administration, and shall have authority and control over all personnel and activities thereof.
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