We don't have ton of information, but we can piece together a few things. Some of your facts are wrong.

First, Ishamael didn't initially needed anyone "attuned" to ta'veren to find Rand, nor was he initially (if ever) using Fain for this. He can find ta'veren in the Pattern himself, like Lanfear he has a talent for this even LTT acknowledged. Ishamael, not Fain, is the one who identified the boys were "the real deal" for the Shadow. Why? Because Jordan explained the boys became ta'veren only shortly before Moiraine showed up, about a year after Fain last visited.

When they did suddenly "flare" in the Pattern, Ishamael found them, and started invading their dreams almost immediately, and had sent a "task force" to pick them up through the Ways. Fain didn't "bring the Trollocs" as Moiraine thought (and beware Fain and his ideas of his own importance...), he was merely there on his routine yearly round of the area as a peddler. Coincidence? Well yes, if you call the Wheel bringing things together around ta'veren coincidences...

Ishamael merely told the Fade to use Fain to identify the right boys at need. This may well have been his big mistake: he delayed by days or weeks the kidnapping operation, to make it coincide with Fain's presence and Bel Tine. Fain himself seems to have been ordered to delay, to justify his presence at the festival (he was expected earlier and would have left earlier too...). Why the eve of Bel Tine? Of course because it's when all the farmers come to the village, and Fain would have revealed to Ishamael that one of the boys of interest was a farmer who lived far. Had Ishamael been willing to make more of a mess, sending the Fade earlier, kidnapping people to learn the location of the three boys named by Fain, he probably would have succeeded. Trying to be "neater" and capture all three at once during a single operation on the eve of Bel Tine, waiting for Fain to arrive to point out the boys and all, gave the Wheel time to have Moiraine, Thom, Lan etc. around and foil Ishamael.

Obviously, Fain was initially one of a great many DF Ishamael had spying around, looking for certain things (which things exactly? We don't know). It just so happens that Fain was one (the only one? we don't know, but almost certainly the only one who did see the "real thing" Ishamael was looking for) who had seen something of interest to Ishamael, who then extirpated from Fain's mind all he had seen about those "three boys" (something like hypnosis using the TP, very likely). It's only then Ishamael did something special to Fain, to focus him on those boys he had seen (it involves a strong compulsion about them, most likely) and seemingly doubling this up with a sensitivity to ta'veren (unless this feature later arose 'accidentally' as a side effect of his compulsion - this is actually what I think happened, as Ishamael didn't need Fain to be ta'veren sensitive, and the boys were not yet ta'veren) - perhaps a routine or "just in case" procedure with DF who saw something of interest, as obviously this wasn't enough for Ishamael to determine this "was it": Fain was just sent back on his normal route, on which he wouldn't see those boys again for a whole year.... The real purpose Fain thus served is to identify for Ishamael there were three boys of interest in EF, three close friends of the same age: a blacksmith apprentice, a shepherd and a do-no-good. It might very well be the inclusion of Perrin in there that attracted Ishamael's interest - because of the dark prophecies. Keeping an eye out for blacksmiths of a certain age is likely one of Ishamael's original orders... Months after he got this information from Fain, the boys suddenly flared as ta'veren overnight and Ishamael located them. Then he knew these were the three boys Fain had seen. He sent the Shadowspawn task force, contacted Fain in his dreams with orders to meet the Fade and spot the boys for him. And we know the rest... Moiraine messed up the operation and the Fade didn't manage to catch up before the party crossed the river. Fain was sent ahead through the Ways to be the spy in Baerlon. Ishamael kept tracking the boys but held off the pursuit. At this point, Ishamael didn't want the attention of the White Tower and Andor, so the Shadowspawn were sent for an ambush in the middle of nowhere, and when it failed it wasn't repeated.

Now, for "what the Shadow knows". You've made a mistake there. You took "what the BA knew" and applied it to the Shadow as a whole. Ishamael wasn't yet around the year Rand was born. The BA acted on their own, torturing Tamra and extracting bits of information from her and deciding to act on them: killing the searchers, starting a hunt for "too lucky men". That was precipitatous, panicky and stupid. Ishamael was around three years later, killed the head of the Black Ajah, forced Alviarin to stop her hunt, and forbid her to take decisions like killing an Amyrlin without his permission.... The BA has been really stupid. Had Ishamael been around, he would have ordered the BA to keep their eyes on Tamra. They would not have found out initially about the searchers, but eventually they would have realised something was up, and spied on those women too.

What information Ishamael has collected over the years, what he's puzzled out, we don't know. But for one thing, he had his knowledge of the KC (much better than anyone else's, as he is an OT native speaker and a writer... subtleties wouldn't escape him)) and he had the Dark Prophecies. Then, very importantly, he knew there was no strong ta'veren curently around.... he knew the DR was to be reborn on Dragonmount, so if he took time to think it through, which the BA didn't, he could very well guess when it was most likely Rand was born: during the battle of the Shining Walls, when Aiel and huge armies were near DM. Ishamael could thus puzzle out his age. It was rather obvious Tamra thought so with her "bounty" idea, the BA just didn't catch that, but it doesn't mean later on Ishamael didn't. He could then puzzle out when the "crucial years" would be: Rand would begin to channel between 18 and 25. Forget the BA hunt for "too lucky men", Ishamael scrapped all this and was in fact furious about it. He reorganized a wholly different hunt later, and we have little idea what his criterias to look for were. Not ta'veren, that's for sure: he would know to look for that, but also not to expect it before the Dragon's quest truly began. He seems to have narrowed down the likely age of Rand. He possibly was looking for blacksmith and apprentices, the one concrete clue the DP gave him (the only one we know, that is...doesn't mean there aren't many more). Aiel colouring might also have been among his criterias - not only because of the timing, but because the fact the old relationship between the Dragon and the Aiel was never a mystery to him.

We also don't know for sure how Moiraine actually found Rand. We don't know for sure if Tam was on her list or not (Jordan decided to remove the name in the novel version of New Spring, a very likely clue he changed his mind and had started developping a more complex backstory around which he would build prequels 2-3) . She probably ran out of names on her list to investigate years before she found Rand. Kari was unlikely to have been in the camps, like common soldiers's spouses. Her family were merchants in Caemlyn, presumably Tam and her lived in Illian for years, and when Tam moved the Companions north for the war, she probably went to her family. Tam was probably alone at DM, and brought back Rand to Kari in Caemlyn. The baby was Aiel, so Tam had all the reasons to keep his finding very secret - Aiel were hated, the baby's safety was in danger. This is in fact why Tam was always so secretive about Rand's origins, and nobody knew he was adopted. Not because he'd been adopted, but because he was an "Aiel savage". It's very unlikely Tam (or even Kari...) went to the Aes Sedai with Rand, for this very reason. It sounds more likely Moiraine heard the story that an officer of the Companions found and saved a child years later, if ever. It seems very unlikely this is what lead Moiraine to the Two Rivers, for the good reason she didn't know she was looking for Tam's child. She didn't know if the boy she had been looking for was Perrin, Mat or Rand - she only found out it was Rand when he began to channel. So it seems she found out information about Tam after coming with the TR, in her old notes or otherwise (Tam was a Blademaster - which Lan would have puzzled out after he got a good look at Rand's sword. it's probably how Lan ended up remembering the name Tamlin al'Thor, a blademaster who lead the Companions back in the Aiel War) and piece it all together with Nynaeve's story that baby Rand was already born when Tam returned with Kari etc. So, why Moiraine ended up in the Two Rivers is still a mystery, but it sounds like she wasn't looking for someone specific. It really sounds like she was following a clue or an intuition about the place, about the Dragon and Manetheren, not tracking down anyone specific from her list there (had she really investigated a lead about Tam and a found baby born on DM, she would have come to the TR long ago, if for nothing else than finding out what his people knew of his current whereabouts after he left the Companions). It also sounds like a Viewing of Min whom Moiraine met in Baerlon was involved - not one that lead to Rand, but which tipped Moiraine that this time was the right one, which lead Moiraine and Lan to rush to the TR. It sounds to me like she decided to start her search in the area in EF itself because it was where Manetheren ended.