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Drow are CE, there is no Drizzt types, Good Drow Goddesses who dance in moonlight, or other crap like that in GH. Again, hard to justify a PC Drow IMO.
Same goes for Monks - typically, the only Monks in Greyhawk were members of the Scarlet Brotherhood, who are LE with plans for world domination. Or at least in my games, that was the case; we had a PC monk once, and he was a turncoat who had escaped the Brotherhood.
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One thing I've never been able to get behind is the idea that a chaotic race of elves would be entirely uniform, embrace monotheism, and abide by Lolth's rules. It appears to me that EGG was of a similar mindset, because whilst Lolth might have lead the dark elves to violently split from their cousins and ultimately slink away beneath the Oerth, they're by no means loyal to her. The most prominent Drow character in the Greyhawk fiction, Eclavdra, was in open rebellion against Lolth, serving Tharizdun and Graz'zt, before being later (post-Gygax) retconned back into a devoted servant of the spider demon. (Maybe I'm wrong about some of those details, if anyone wants to correct me please don't hesitate. It's... been a while.)
In my game, there are enclaves of dark elves, and at least one dark elf city, which openly reject Lolth and the Drow that follow her. (All Drow are dark elves, not all dark elves are Drow.) If it is possible for players to make characters that are valley elves (like, oh my gawd) or Grugach, then it shouldn't be out of the question to roll up a dark elf. It would probably be necessary to take proficiency in the disguise kit, though... dark elves would be "kill on sight" to surface elves, Drow, and most human and demihuman populations.
It seems to be somewhat accepted for Vhaeraun to exist in Greyspace, so having some version of Eilistraee isn't problematic at all, especially as I conceive of the two. I imagine these two (the twins) as different sides of the same coin, to the point that worship of either one "feeds" both. Neither can exist (or be destroyed) without the other. I wouldn't have her function as a goddess of good drow (because there are no good Drow,) but she could grant spells to dark elves either as a minor power (equal to her brother) or as a warlock patron.
With regard to monks, I think limiting them entirely to the Scarlet Brotherhood defeats what I see as the core narrative of the class: power from within. The wizard channels cosmic power, the cleric channels divine mojo, the sorcerer taps into some empowered heritage... but the monk looks within to find the power within all beings, and expresses it. While it might be most common for a monk to be trained in an abbey or monastery, I see no reason why a PC monk couldn't simply be a goatherd with a penchant for self reflection, with maybe a folk hero background.
With regard to tieflings, aasimar, and dragonborn, I agree that these aren't races or populations in Greyhawk, but rather individual halfbreeds resulting from intimate encounters (probably accomplished via shapechange) with fiends, celestials, or dragons. You don't have to eliminate these races to make the setting distinct, just make them freaks. The same is true of warforged, for that matter: there's a character in a Greyhawk based campaign I'm running right now who was an archmage's champion and fell in battle in such a way that "normal" resurrection wasn't an option (long story,) so a construct body was crafted and his soul/spirit invested therein. The traumatic experience and unfamiliar body effectively knock him back to level 1. Viola, warforged PC.
The virtue of the World of Greyhawk is not found in what it doesn't have - you can have any damn thing you want in a Greyhawk campaign. The virtue lies in the presentation and tone.