Can a paladin use the 1st level spell from the Magic Initiate feat, or the spells gained from a race (like Tiefling, or Eladrin) to smite?
The Paladin requires spell slots to smite. As referenced from the PHB Errata:
"Divine Smite (p. 85). You can expend any spell slot, not just a paladin spell slot."
This means that things like Warlock spell slots do work for Smite (even though they recharge after a short rest), whereas abilities that let you cast spells without slots (e.g. most racial abilities and Magic Initiate) do not.
Really it depends on your dungeon master. I would probably allow it, since usually those spells are more helpful or chosen for specific reasons than even class spells. For example, a tiefling expending a hellish rebuke for a smite looses the ability to punish melee attackers for decent fire damage, so yes, I would allow it, but only as a base level smite, in other words, regardless of the spell's actual level it counts as a first level slot for smiting. However, according to firm rules, no, you cannot, as stated above. An alternative for a DM would be to allow you to expend, say, Hellish Rebuke to smite, but the smite would be fire based not radiant, a reflection of your Tiefling heritage. But without consulting with your DM I would not count on it for build purposes.