Nah, that game was just a metaphor, not a real thing--as shown by many scenes which would be entirely illogical otherwise (or from the fact that the entirety of those novels wouldn't have mattered even within their own context if that game was real, since the game would be the only real determining factor, and stuff done by mortal would be only fluff). It would also be incredibly OOC for Eilistraee to even think taking part in a thing that uses people as literal pawns. Then again, being absolute nonsense is the trademark of that series. It goes to the point that it couldn't have even been written if not for the authors warping characters and lore, because they based the whole plot on false information (like the position of Eilistraee's realm) and certain characters doing things that they would have never done without the warping they did (their smearing of Eilistraee is notorious--and intentional--for example, down to sniping specific lines of lore about her just so subvert them in VERY offputting ways). It's one of the reasons why their retcon makes much more sense than the novels themselves.
As for Corellon, I doubt he'd even care going after Lolth. Even in the novels He merely collected a handful free souls, and was willing to even let the majority of Eilistraee's followers to rot, because they weren't affected by a race change (yes, only a few hudnreds of the thousands of Eilistraee's followers were transformed by that mage, which is why the transformation into dark elves did absoilutely nothing in the big picture). He also proceeded to do absolutely nothing for a century or so after that event. Then again, setting aside WotC's retcon, if we want to analyze the situation, since Ed Greenwood explained that Eilistraee survived with the help of Mystra (and so did Vhaeraun, whom she spared but whose portfolio she temporarily took), it could easily be that Corellon only subentered to "watch over" his daughter's followers while she was powerless as a deity and was recovering. This would paint a much better picture of him thatn the novels do, tbh, and given that Ed spilled the beans about the "behind the scenes" of the events, I'd say this is the most likely possibility.
That said, the idea that the drow would automatically worship the deity who kills Lolth is flawed IMHO. People don't work like that; the Lolthite drow would likely just descend into a messy massacre to determine who would fill the power vacuum left by the now-powerless priesteses (and they would do all they can to hide their lack of power, which could include turning to demon worship to get powers and stuff). Little would change for the other drow factions, aside from an opportunity for the likes of Vhaeraun, since he'd have a good chance at establishing dominance, or at least a much easier time destabilizing and demolishing the Lolthite society in the future. For the Eilistraeans, there would be A LOT of work to try and rescue many of the victims of the massive bloodshed that would ensue. Other races, like the dwarves, if they ever came to know about Lolth's death, might use the civil war of her followers as an opportunity to get rid of drow cities near to them.
I wouldn't say it's unlikely. Lolth made a really stupid move by isolating her own Realm from the Abyss. If Lolth didn't have plot armor, since she's alone, since she's made enemies out of everyone, chances are we would see stuff like the Mordinsamman and the Seldarine doing a blitz in the Demonwibs, erasing her from the Realms. I mean, Vhaeraun could even be convinced to join, and Eilistraee wouldn't certainly oppose them. What would remain as her ally? Selvetarm? Moradin's laugh would be enough to send him flying into the horizon. Not to mention that even Selvetarm hates her, thiugh he still defends her for whatever reason.If some other, non-elven deity managed to kill Lolth outside of the sava context (which would be challenging outside of some kind of cataclysmic event that rendered the gods vulnerable like during the Time of Troubles), they would absorb Lolth's portfolio and would probably continue to pretend to be Lolth because they'd gain the greatest support from the Drow in that way and only very gradually alter drow society to suit their desires.
Lolth is one of those villains that only survives because of author bias, really. If not for her adversaries getting her, it should've been a lot of the drow losing faith or bringing change over friggin' 12000 years of utter misery (both emotional and material), stagnation, and regress from their golden age from before Lolth's cult was even a thing. Don't give me the " but a literal deity keeps them down..." justification; it's weak. Threats of death and massacres never stopped IRL humans (who had MUCH better conditions than the vast majority of drow, who can literally be killed or bankrupted and then enslaved for fun in Lolthite society) from rebelling or even displaying dissent. And in the Realms even internal disillusion, without the need to display it outside, without the need to do *anything* material, would be enough to kill or at least demolish Lolth's power, due to how deities work in FR (and that was true even before the ToT, as shown by many deities fading due to loss of worshippers). Now, add opther deities who go out of their way to help the drow break free, and you have a picture where the current canon is ridiculous.
As for the portfolio, it wouldn't necessarily go to the killer. There are cases of deities being killed and their porfolio remaining unfulfilled. Mystra, for example--though we know she had contingencies and survived, she effectively stopped functioning as a deity even thogh she was still alive. Same for the Dark Seldarine.