Can I buy stock in any of those government agencies? No...but I can buy stock in the FRB although I would have to own a bank to do so.
People that work for these independent agencies of the Executive branch are federal employees and their paychecks and terms of service are all administered (just like the military, federal judges, the FBI, etc.) by the Office of Personnel Management --another independent agency.
I've provided links from a Federal reserve bank stating very clearly that employees of the federal reserve bank are not government employees. I've also provided links that federal courts have ruled that the FRB is not a government agency and people employed by them are not government employees. Not to mention that I've also linked an OPM document that lists all federal government agencies and the FRB is absent. I don't know what else to tell you.
BTW The FRB employees are not paid through the OPM but directly by the FRB out of “operating costs”.
What's so great about Fed stock?
Let's get clear on the difference between the Federal Reserve System where Bernanke works, and the district banks that --
...operate under the general supervision of the Board of Governors in Washington. Each Bank has a nine-member Board of Directors that oversees its operations.
Federal Reserve Banks generate their own income, primarily from interest earned on government securities that are acquired in the course of Federal Reserve monetary policy actions. A secondary source of income is derived from the provision of priced services to depository institutions, as required by the Monetary Control Act of 1980...
The article's about the Federal Reserve System, not the banks. In Washington DC there are about a thousand federal government employees that don't work for district banks. There's no private stock ownership of the Federal Reserve System that we're talking about. This is why you may want to reconsider what you said in your post #3.