Does the Federal Reserve buy and sell stocks? Can the Fed buy and sell stock in publicly traded companies? Is there evidence of this and, wouldn't this behavior drive the price as opposed to actual market forces?
No, the Fed is not allowed to buy stocks, they are allowed to buy government securities in open market operations in order to achieve the target rate for the federal funds rate. The guidelines for this are explained in the Section 14 of the Federal Reserve Act. You can find the Fed holdings in the Federal Reserve Statistics.

However other central banks, like the Bank of Japan, started buying stocks as a measure to support their financial institutions (their banks were subject to too much market risk because of their stock holdings). The have detailed their stock purchasing plan in their website.