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).

You could use both present and past in these examples. However you should consider the following: Reported speech: If you are reporting what someone said, then you can use past tense. However you can also use present tense. When something is always true (like name and earth) you would use 'is' present tense because these things don't change and they are true NOW. A name never changes. The earth being round never changes. Also in example 2 present tense is also correct if the person is alive still today (now). If the person is dead, you would use past tense.

Exactly. Things are not always clear-cut. In examples like this, you can use either 'is' or 'was'. In fact, in these particular sentences, I think that 'was' sounds more natural, as it follows the internal sequence of tenses. In a Cambridge Use of English exam, you would get either one marked correct.