[RPG] If you get a spell through a racial ability, can you also take it as a spell from a class

No. Spells Known is a part of the Spellcasting or Pact Magic features. Any spells granted from other features do not count against this limit, unless explicitly stated.

Yes! A drow's darkness spell does not use spell slots (PHB, p. 24, emphasis mine):

When you reach 5th level, you can also cast the darkness spell once per long rest.

That's easy enough to overlook when learning the game, since it's just four words, and it isn't obvious how this is different from normal spellcasting until you're quite familiar with how the game works.

The upshot of this is that you don't need to spend any slots to cast the spell. You don't even need to be a spellcaster class. The disadvantage is that, since it doesn't use slots at all, you can't ever cast it more than once per day.

(To cast it more than once per day, you'd have to learn/access the spell as a Eldritch Knight or Cleric, and cast it as a normal spell. Then you'd have it via spell slots, and an extra once-per-day racial casting that didn't use slots at all.)