Nah, the idea that a god has to be omnipotent and all powerful is actually pretty new, and localized primarily to the Jewish->Christian->Muslim faiths, and the few alternative monotheists. If you look at the a lot of the older religions, particularly the pantheist belief systems (multiple gods), you realize that they are all very similar to the Warhammer "gods". The "big three" of Classical History would be the Greek/Roman Pantheon, and the Celtic/Nordic pantheon. When you look at gods like Zeuss/Jupiter or Thor, they are "king gods" and still aren't omnipotent. The other gods can play tricks upon them, and there are limitations to their powers, even over the mortal world.
Zoroastrianism was the root of monotheistic religions, as it was the inspiration for our later single-God, good vs evil, omnipotent religions (ie: Judaism, Christianity, Islam). Prior to that, at least in Western religions, there was much more focus on religions where "Gods as celebrities", and had more limitations akin to your modern day superhero than the omnipotent "God".
In a way, Western religions had "things they were in charge of", but were fairly open to what they could actually do. For example, Ares might have been the God of War, but it's not as though anybody couldn't have picked up a spear and gone on a stabbing spree. He didn't have any special "War Powers", he had powers that all gods had (turn into critters, fast travel, immortality but not invincibility, etc), he just had a different job than others had.
Ultimately, though, I think we're just splitting hairs. At what point of "being able to manifest creations on a whim" and "immortal creature" and "lives on its own plane of existance" does one go from being a god, to being God? Really, the separation just seems to be whether one part of a polytheistic pantheon or monotheistic one.
Also, the Warp always existed, and the Eldar Gods did indeed have ties to it. I'm not positive how the Old Ones factor in to 40k lore, but I know that in Fantasy, they are the ones who opened the rift between the Warp and Reality. For the Eldar Gods, the hubris of the Eldar eventually led to the birth of Slaanesh - it can be assumed that the Eldar gods were created the same way, that Slaanesh is essentially the Eldar god of excess.
I have always viewed the 40k and Fantasy mythology to be one of faith. In Fantasy, Sigmar = Emperor. He was "just a guy" at the start of his life, but rises to become something more because eventually the entire realm came to believe that he was more than human. Where the Chaos Gods thrive on lack of faith - fueled by raw emotion and base desires - the other gods are created because enough people believe in them. Very similar to Douglas Adams' "Hitchhikers Trilogoy", when he explains that God died in a puff of reason once the world collectively stopped believing in him.
Part of the problem with 40k origin stories, is that it's hard to tell what is still valid canon, and what has been removed over the years.
But, I'm trying to remember if 40k really ever discusses how the Old Ones came across their connection to the Warp. It's just a given that the Old Ones use the Warp as their thing, it's their shtick, along with all their progeny. But they treat the Warp as a tool to be used, similar to how the Eldar would treat the Warp if they did not have to concern themselves with She Who Thirsts. In the end, though, the Warp also is the cause of their downfall (Old Ones and their creations suffered from the Enslaver plague, but not non-Warp creatures like the Necrons).
The 40k pantheon works out really wierd, in that it's not cut and dry that they all hung out in the Warp. In fact, I'd argue that Eldar gods never did, and would think that the Warp was always a "bad place". Case in point, though somewhat outdated now: old fluff talks about the Eldar gods having interactions with the C'Tan named the Deceiver, but in old fluff the Warp was anathema to the C'Tan and the C'Tan could no travel there.
Or perhaps the Eldar gods simply traveled between dimensions with much more freedom than any others. Afterall, the C'Tan were restricted to the material realm, and never had the option to travel to the Warp. On the other side of the coin, the Chaos gods are also restricted to the Warp, where they exist as both rather nebulous concepts, as well as persons in their own right. Yet Khorne doesn't really make appearances to the material realm. He doesn't take vacations to Mars. Instead, portions of his personality are manifested to the material realm as lesser daemons (well, lesser than Khorne). So, the way the two fought between themselves was not directly. There were no slug-fest between the Nightbringer and Khorne, because that was impossible. So, instead, the C'Tan (in old fluff) "fought" against the Chaos gods through attempts to close off the Warp.
So, in that way, Eldar gods were "their own thing", as different from the Chaos gods as the Chaos gods were different from the C'Tan because they either seemed to travel between the realms freely or (and this is my theory), they existed in their own place. And of course, and I cannot emphasize this enough, that's even
if the Eldar gods existed as actual "gods" at all.
Part of the Eldar thing (being elves and such) is that they are supposed to talk entirely in metaphors and poetry. So, Eldar gods might not have ever been actual beings, but simply stories and parables that occurred to the Eldar race manifested as anthropomorphic entities. Interpreting Eldar stories to be literally talking about "gods" might just be our human interpretation.
As for the Emperor: while he was never as powerful as he was during his 30k period, the guy was always the mover and shaker of the world, changing the paths of destiny every time he surfaced. It rather changes depending on the story, but sometimes he's immortal and sometimes he's mortal and resurrects here and there. But his power does not manifest from the faith of his followers; he was most powerful at a time when he was discouraging such worship among humankind.
Really, only the Warp and its creatures are fueled by following and faith.
