Learn Croatian: Dostavljamo humanitarnu pomoć diljem ugroženog područja. - We are delivering humanitarian aid across the endangered area.

Yes, širom ugroženog područja would mean exactly the same thing. Both diljem and širom are used to express spatial distribution ('across' or 'throughout') and both require the genitive case.

Why does pomoć look like it is in the nominative case when it acts as the direct object?

Pomoć is actually in the accusative case here, acting as the direct object of dostavljamo. It is a feminine noun ending in a consonant (an 'i-declension' noun), and for these nouns, the accusative singular form is identical to the nominative. You can tell it is accusative because the adjective modifying it, humanitarnu, takes the regular feminine accusative ending -u.

Is ugroženog a regular adjective?

It acts as an adjective here, but it is originally the passive past participle of the verb ugroziti (to endanger or threaten). It takes the neuter genitive ending -og to agree with the noun područja.