I can’t find any suggestion that the giants had anything to do with the origin of the elves; the elves seem to predate the giants’ rise to power, actually existing during the Age of Demons, at which time the “lesser races” (i.e. not demon or dragon, apparently including at least the elves, giants, goblins, and orcs) simply tried to avoid getting killed in the war between demons and dragons.
After the couatls sacrificed themselves to end that Age, the giants were able to rise to power in Xen’drik using dragon-taught magic, ushering the Age of Giants.
The giant empire then enslaved elves, and thus “inadvertently” raised them up out of their primitive state, teaching them magic and the like. Towards the end of, or just after, the war with the quori, the elves revolted. The giants were going to put this down using the same cataclysmic magic they’d used against the quori, but the dragons came and crushed them, instead. An elven prophet/hero named Aeren, among other things, foresaw this end and signaled that the elves should flee Xen’drik before it happened.
Those elves named the island they fled to “Aeren’s Rest” or Aerenal, and referred to themselves as the Aereni. Aeren had sacrificed himself during the exodus, but not before figuring out how to do the whole undying thing, so he became the first undying councilor.
The drow, of course, stayed in Xen’drik. Simply based on the fact that all elves from Xen’drik are drow and no Aerenal elves are drow, it seems likely that the actual distinction between “elf” and “drow” happened after the exodus, probably during or immediately after during the dragonfire-purge.
However, Secrets of Xen’drik does have this:
According to one legend, the Sul’at giants created the drow by binding dark forces to normal elves
The Sul’at League was a series of fire giant city-states, founded by the first fire giant Adaxus, who got that way through some fiendish pact. Or so legend has it, anyway. This is the only mention I can find of giants manipulating an elven race. That said, it doesn’t seem to jive with the facts, which calls that legend into doubt. It seems unlikely that every single drow would choose to stay and every single elf would choose to leave.
The elven trance is different from sleep, and their “dreams” are different as well. I am still searching for information about whether or not these dreams are similar enough that they still go to Dal’Quor in spirit while they trance.
Unfortunately, I have so far looked through Eberron Campaign Setting, Secrets of Xen’drik, Secrets of Sarlona, Magic of Eberron, Races of Eberron, and Player’s Guide to Eberron, all without finding any reference to elven dreaming or lack thereof, much less whether or not the giants were responsible for it. I didn’t read those books cover-to-cover, but anything that sounded like it might even possibly have the reference, I checked. I’m running out of even remotely likely suspects.