This part of your question is pretty well covered in New Spring. Moiraine and Siuan had books of names and locations for all the children born during the time of Gitara's foretelling. Obviously Tam wasn't likely to have signed up for the Bounty Tamara was giving out, but it seems probable that one of the women in his camp could have passed at least partial information along to Moiraine between the events of NS and EoTW.
Rand would not have been on the list. The list did not contain the name of the Aiel baby born at the right time near Dragonmount. The list was very long. Groups of 2 Accepted were sent out from sun up to sundown taking names for days or weeks. Moiraine and Lan were searching by themselves which would have taken years. Someone would have had to remember Tam finding a baby after 20 years.
The pattern willed it. Otherwise we wouldn't have this terrific story to read and comment about.
Actually Rand was on this list I can't remember what page it was, but she actually mentions rand tam and Kara, saying how aggravated she was over the listing because all it said was something like rand born to tam and Kara by dragon mount, in new spring. Most likely someone else under tams command and in tams army reported rand being born to Tam and Kara, he is listed as their son. If i find the page number I'll psot it for the paper back version but i really don't remember were it was in the book and don't feel like looking through the whole thing again.
Reread - Moraine, Lan, Myrelle, and Nyneave
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