This is an interesting case. It’s correct either way, but if the fact were still known to be true (or at least from a reliable source), I would tend not to backshift the tense: "The teacher said the earth is round."
In your example number 3, however, the fact is false and this has come to my attention since then, so I would tend to use the past tense to refer to the false information that was communicated in the past.
For your example number 4, I would once again tend to use the past tense, since the my past uncertainty has now been resolved.
For examples 1 and 2, I would definitely use the past tense. He may have changed his name. As for the hero, 9-11 is in the past; so in a sense is her heroism. Also, she may well be dead now, in which case she isn’t anything (other than dead and buried).