In the United States, can private companies issue publicly-traded bonds?
If so, are these bonds usually available to retail investors, or are they restricted to accredited investors?
Where do these bond issuers publicly file their financial information (e.g., which SEC forms do they file)?
Absolutely. For example, let's look at two of the largest US private companies: Cargill and Koch Industries.
You can see the rating for a bond issued by Cargill: a USD 50 million issue paying a 7.1% coupon, maturing 20270729, with ISIN US141784AR94. You can see the latest Moody's review of Koch Industries credit and a rating of their commercial paper (short term debt). The same is true internationally: you can read about the secretive private firm Glencore selling bonds.
Companies do have to divulge a little bit more information when they offer bonds to the public. The article here, on Citadel's first bond issue, revealed information on funds which could be withdrawn that was previously not widely known.
You do not need to be an accredited investor to invest in these bonds since they are filing reports with the SEC. The firms file forms 10-K and 10-Q, however those filings may not be public. You can read Helwege and Packer (2009) if you want to see a study that discusses a dataset built from such filings.