No, you can't cast it using a spell slot.

Spells you can cast because of your race aren't spells you know, and thus can't be spells you prepare. Per the PHB (p. 201, "Known and Prepared Spells"):

Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind ...

... and it goes on to describe some of the class-specific variations of knowing and preparing spells. But in any case, having a feature from your race or class that allows you to cast a spell isn't the same as knowing the spell.

This is spelled out clearly in the multiclassing rules for Pact Magic quoted in the question (PHB p. 164; emphasis mine).

If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.

Likewise, the Drow Magic trait clearly distinguishes the known cantrip from the merely castable other spells (PHB, p. 24; bold for emphasis mine):

You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.

If the intent of the rules was that Drow also knew the non-cantrip spells, it would say so.

Eldritch Smite is Warlock slot only

Only warlock spell slots from the Pact Magic feature can be expended to use Eldritch Smite; spell slots from the Spellcasting feature do not work with it.

An unofficial tweet by rules designer Jeremy Crawford in November 2017 confirms that this was intentional (unlike the paladin's Divine Smite, which had similar wording initially but was changed in errata later to work with any spell slot):

Talking about Eldritch Smite; can you only use Warlock spell slots to power it, or can you use any slot (à la Divine Smite)? The RAW on it specifies Warlock slots.

Eldritch Smite works with warlock spell slots only—the ones you get from Pact Magic.