I allow multiclassing if it makes sense lore-wise and world-wise. For example, a Cleric/Warlock is fine, as long as the warlock patron is a servant or an ally of you deity, and if you can come up with a story to justify it. There could also be the whole "an evil entity is trying to claim the character's soul, but the character is trying to escape the consequences of their pact/the entity's infuence by latching themselves onto a good deity" thingy. However, barring extremely pragmatic gods, that stroy would beg the question of why the character is even continuing to channel the evil entity's power.
Same thing for a paladin/warlock--your patron and your beliefs have to align at very least. A Bard/Cleric would be awesome for priest(esse)s of deities of arts, beauty, music, etc... Combining classes like Fighter, Rogue, and Monk could even simply represent the character experimenting with different combat tactics in their career.
Of course, picking a class mid-game would require a proper story/introduction to justify that. No gaining powers out of the blue, it's just jarring.
The sun is a very threat to your existence. It wouldn't feel good.
But that is beside the point, why a Drow? Why not any of a myriad of other races?
1. To be the centre of attention.
2. To tell a story to a captive audience rather than to create a story with friends.
It's just disrespectful. At our table we don't allow any of that sort of thing. You're here to contribute to the group or you should go write a novel on your own.
Some worlds do have drow presence on the surface--the Realms included, with 2 whole drow deities and their faiths dedicated to that, so it could very well make total sense, lore-wise (I mean, the possibility exists that they were born on the surface, and that has been true for millennia, in-universe). Some of those drow have managed to gain some presence in other races' settlements, like Waterdeep. On a side note, in FR at least, the drow only take as little as 10 years to adapt to sunlight. They're no vampires.
That said, the reason for picking a drow could simply be that the person likes the concept, which can be spun in many ways. The very part of conquering a new beginning (if the player is even going for a drow that came from a Lolthite background) can be played in many different ways, without begging for attention or anything like that--just like a character of any other race could have a story about rising from a shady past.
If their very presence is going to be problematic, disguises and magic initiate with disguise self (or picking that spell if you're a caster with it in your list) can go a long way. I mean, why assume that someone is going to be a drama queen when they could just be invested into the character?