True, although when one part of the state prints money to "lend" to another part of the state, I'm not sure how useful it is to include that as part of a total "debt" figure. It's just an artefact of the way the US chooses to implement its monetary policy.
The Fed is a private institution, though a small stake is owned by the US government. So it's not really one part of the state lending it to another part of the state.
"It is not 'owned' by anyone and is 'not a private, profit-making institution'. Instead, it is an independent entity within the government," however for our purposes this quote is probably more relevant, "the Federal Reserve must work within the framework of the overall objectives of economic and financial policy established by the government."
So, independent, but founded to enact government policy.
While your summary is technically correct, just look at any of your bills. What does it say? "Federal Reserve Note". Your money is not printed by your government. With all the implications that carries, including your government lending it from FED on interest.
The majority is owned by US institutions (mostly the fed):
http://en.wikipedia.org/wiki/File:Estimated_ownership_of_tre...