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.

Yes it is viable but not with Shield.

Your best bet would to use Mage Armor, since it is a single action spell as opposed to the reaction of Shield that has specifics on when you are allowed to cast it (emphasis mine).

Casting Time: 1 reaction, which you take when you are hit by an attack or targeted by the magic missile spell

So unless you actually have someone hit you repeatedly you can't cast this spell all that often per PHB 196. At least by RAW.

A reaction is an instant response to a trigger of some kind...

This section goes on to say that the opportunity attack is one such instance of a reaction but you can't take it unless something would trigger that as detailed in all the descriptions of reactions. This why when you ready an action you are required to specify a trigger.

Mage Armor could be used as often as necessary to regenerate the ward, unless there is errata somewhere that changes the wording to slots as opposed to level. One thing to note is that spells with verbal components could be overheard and thus could give away position or spoil the element of surprise, so choose when to do this with care.