When the Fed lends money to banks, you know they aren't actually lending money, right?

They're creating high-powered money, against which the banks can create money by loaning it into existence. Banks can also get a nice carry trade.

How do commercial banks lend money to the Treasury?

Through buying Treasuries. The Fed can't purchase them directly; it needs to go through primary dealers. In theory, the Fed isn't supposed to bankroll the Treasury, but that's theory. In practice, it's currently buying more than half of all new debt issued by the Treasury. But Europe is farther along the road to crisis than us, so nobody really cares in the short term.

Regarding GDP, I don't think it's a fixed quantity at all! The issue is more that government spending is taking away from what entrepreneurs and individuals would spend and replacing it with what Gosplan wants, which doesn't exactly respond to market signals. When government spends money, it's denying private individuals the chance to save or spend that money to satisfy their desires. Not that ZIRP isn't already killing saving by holding the price of money artificially low.

It's tricky talking about inflation because there are both monetary and price aspects at work. One can talk about inflation and deflation of the money supply. These are both linked to to price appreciation and depreciation, but like you point out, other factors are at work.