The Federal Reserve doesn’t auction T-Bills, it’s the Treasury that conducts T-Bill auctions. The Fed buys and sells T-Bills in open market operations, much like any other other buyer and seller.
The Fed doesn’t use physical money, it uses ledger entries, computer data entries, that sort of thing. When the Fed purchases a T-Bill it credits the dollar amount of the T-Bill it purchases to whomever sold the T-Bill to the Fed.
What the Fed is doing is altering the mix of money. Dollars exist as illiquid Treasury paper, or as “high-powered money”, which is currency or checking deposits, that sort of thing. The Fed “creates money” when it purchases a T-Bill, it reduces the money supply when it sells a T-Bill.
What will determine inflation is a combination of the quantity of high powered money, and how fast that money changes hands- the “velocity of money”. Right now the velocity of money is quite low. And while the quantity of money appears to be growing dramatically, that may not be entirely accurate, as there are some strong deflationary forces at work when loans default.
When the Fed begins to detect inflation they will start selling their stock of T-Bills. When the Fed sells T-Bills it removes high powered money from the banking system in exchange for illiquid T-Bills. You’ll hear this described as “the Fed soaking up liquidity”.
If the Fed’s actions were balanced the value of the dollar wouldn’t be affected over the long term. Actually the dollar has lost 97% of its value since the Fed began to “stabilize” the economy.