Learn Russian: Стоило дежурному позвонить, как пожарный выбежал, надевая шлем. - As soon as the duty officer called, the firefighter ran out, putting on his helmet.

This is required by the стоило ... как construction, which means 'as soon as' or 'no sooner had'. In this structure, the person performing the first action is the logical subject and must take the dative case. The action itself stays in the infinitive (позвонить).

Could we use когда instead of как in this sentence?

Not with стоило. The construction стоило (plus dative and infinitive) strictly pairs with как to mean 'as soon as X happened, Y immediately followed'. Using когда here would be a grammatical mistake.

Are дежурный and пожарный nouns or adjectives?

They are adjectives that function entirely as nouns (substantivized adjectives). They mean 'duty officer' (literally 'the on-duty one') and 'firefighter'. Because of their forms, they decline exactly like standard adjectives, which is why we get the dative adjective ending -ому in дежурному.

What form is надевая and why use it here?

Надевая is an imperfective verbal adverb (sometimes called a gerund). It describes a background action happening at the exact same time as the main action (выбежал). It efficiently translates to 'putting on' or 'while putting on' without needing a whole new clause.