Does each god know which god is which? And can I ask the same question twice to the same god?
Yes, True and False have to be omniscient to be able to answer consistently correctly or incorrectly, for any arbitrary binary question. There's a version of the answer which (spoiler) relies on asking unanswerable questions, which only Random would answer. There's also solution that doesn't rely on such gimmicks, however.
P/S/A: There are single sentences which can create life-changing amounts of difference.