Learn German: Wir vertrauen dem Zwerg als unserem Heiler.

The word als (as) links two things and makes them share the same case. Because unserem Heiler describes the role of dem Zwerg, and dem Zwerg is in the dative case, the phrase after als must match it and also be in the dative case.

Could I just say Wir vertrauen dem Zwerg als unser Heiler?

No, that would sound wrong in German. If you used the nominative unser Heiler, it would grammatically refer back to the subject of the sentence, wir (we). Because you are saying the dwarf is the healer, the cases of those two nouns must match perfectly.

Why is dem Zwerg in the dative case to begin with?

The verb vertrauen (to trust) is a "dative verb." Even though the dwarf feels like a direct object (which is usually accusative), vertrauen always requires its object to be in the dative case.

Where does the -em ending on unserem come from?

It is the standard dative ending for masculine words. Heiler is a masculine noun (der Heiler). Since the phrase must be in the dative case, the possessive word unser (our) gets the masculine dative ending -em, becoming unserem.