hawkeyefan
Legend
This is sort of assuming a certain level of the specificity on the part of what the character wants to do and no more.
"Let's design some evil Drow civilisation" assumes that the GM is still making up the Drow civilisation - but in my experience specificity often breeds more specificity. "I want to play a Drow and I thought maybe could have been a former member of the first half of the city but his house lost favour with the Spider Queen and was destroyed when the other houses launched an attack on each other and he was forced to flee the city". This, assuming you go with it, would impose a the default Forgotten Realms conception of Drow onto the game.
Possibly, yes. I went with the most common reason I've seen that players have wanted to play drow, and once that was assumed, I came up with some basics about how that would come about. But I never said anything about Lolth or matriarchy or the underdark. I took the broad theme and then said "this is what the world will need in order for you to have your character theme".
If instead, the player just wanted to play a drow as a refugee from the underdark who's a stranger to the surface world, then you'd need to come up with the underdark or something similar.
But yes, once you come up with whatever basics are needed, then usually additional details will suggest themselves, and the GM and players can come up with those ahead of time, or leave it to be determined in play.