Can bilirubin from a donors blood affect the recipient, since checking for bilirubin levels is not part of the criteria for donation?

Sort of: The transfusion of blood causes some collateral damage to at least some of the rbcs. Red cell damage/destruction produces an elevation in bilirubin. Not "from the donor" in a pure sense.

Not tested: Bilirubin is normal in every person and should not affect the recipient of blood transfusions. Bilirubin levels may be elevated due to underlying disease process, in which case the potential donor should have signs and symptoms making him or her ineligible for donation during the medical screen and physical exam before donation.