The answer to the first question is ‘sort of'. The answer to the second question is no. The effects of having an institution with the power to manipulate interest rates and the money supply at whim are equally pernicious whether the institution is privately or publicly owned. However, if you strongly believe that the government can not only be trusted to ‘manage' money and interest rates but is capable of doing so to the benefit of the economy, then please contact me immediately because I can do you a terrific deal on the purchase of the Eiffel Tower.
The fact is that the federal reserve would be a really bad idea regardless of whether it were privately owned or owned by the US government. The question of ownership is therefore secondary and the people who stridently complain about the Fed being privately owned are missing the critical point. In any case and as I explained in an article way back in 2007, the Fed is not privately owned in the true meaning of the word “owned”. For all intents and purposes, it is an agency of the US Federal Government.
In addition to the work of G. Edward Griffin referenced in my above-linked 2007 article, useful information about the Fed's ownership can be found in a 2010 article posted at the Mises.org web site. This article approaches the Fed's ownership and control from an accounting perspective, that is, by applying Generally Accepted Accounting Principles (GAAP), and concludes that:
“…the Fed, when tested against GAAP as the Fed itself uses it in the Fed's assessments of those it regulates, is a Special Purpose Entity of the federal government (or, according to the latest definition, is a Variable Interest Entity of the federal government). The rules of consolidation therefore apply, and the Fed must be seen as controlled by federal government, making it indivisibly part of the federal government. The pretence of independence is no more than that, a pretence.