For example a Drow has the racial spell Darkness, spell level 2 (gained at player level 5). Can he cast the spell even if he is multiclassed (Eldritch Knight 3; Cleric 2) and only has spell slots up to level 1? The spell requires spell slot 2 and that the player be level 5 total, so if he wants to cast it does he have to wait until he has a 2nd level spell slot?
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.)