My question concerns the total assets of the Federal Reserve. To what extent does the government have control over, or own these assets?
The U.S. government only owns the assets deposited in its own accounts in Federal Reserve banks and profits from portions of its operations amounting to about $100 billion a year.
Federal Reserve member banks are owned by private commercial banks in their region and are partially controlled by these shareholders (who have some say in electing regional Federal Reserve bank boards but not in the manner of a typical privately held company), but certain national economic policy issues delegated to the Federal Reserve's policy making entities are controlled by a Presidentially appointed (and U.S. Senate approved) Board of Directors which is really what people are talking about when they refer to the "Fed".
The overall structure is a complex public-private hybrid organization and this brief answer oversimplifies the matter which has many nuances and unique features.
Most notably, the Federal Reserve's policy making board sets some key interest rates, sets the reserve requirements for commercial banks (together with and in coordination with the FDIC) and as a matter of practical reality, is the primary entity that regulates the money supply. The Federal Reserve's policy making arm can also authorize loans of U.S. funds to private business in certain circumstances (most prominently in recent times during the Financial Crisis).
The Federal Reserve is also the primary administrator of the U.S. payments systems used for checks and similar negotiable instruments, and for wire transfers between financial institutions.