Learn Hardcore German: Auf dem Bürgersteig warteten die Passagiere in aller Eile auf den nächsten gelben Bus. - On the sidewalk the passengers waited in great haste for the next yellow bus.

auf dem Bürgersteig uses the dative case because auf here expresses a static location (“on the sidewalk”) and with two-way prepositions you use dative for location.
auf den nächsten gelben Bus uses the accusative case because warten auf is a verb + preposition combination that requires its object in the accusative (“to wait for …”).

Why is in aller Eile genitive instead of dative, and what does it mean?

in aller Eile is a fixed idiomatic expression meaning “in great haste.” It uses the genitive singular of Eile (die Eile → der Eile), intensified by the pronoun aller. You could say in Eile (“in haste”), but in aller Eile is stronger. Saying in der Eile or just eilig would change the nuance or require a different construction (e.g. sehr eilig).

Both nächsten and gelben end in -en. Why do these adjectives take that ending?

They follow a weak declension because they come after the definite article den. In masculine accusative singular, all weak-declension adjectives end in -en: • der junge Mann (nom.) → den jungen Mann (acc.)
So here: den + nächsten + gelben + Bus, all with -en.

Why is the adjective order nächsten gelben Bus and not gelben nächsten Bus?

German generally orders multiple adjectives by type. Ordinal or numeral adjectives (like nächsten) come before descriptive adjectives (like gelben). So you get “next (ordinal) yellow (color) bus.”

Why does the finite verb warteten appear in second position, and why is die Passagiere after it?

In main clauses German follows the verb-second (V2) rule. You may begin with any one element (here Auf dem Bürgersteig), but the conjugated verb must occupy the second position. The subject often follows the verb if something else is topicalized in first position.

What is the function of auf den nächsten gelben Bus in the sentence, and how is it different from auf dem Bürgersteig?

auf den nächsten gelben Bus is a prepositional object required by the verb warten (warten auf + Acc.), acting like a direct object with a preposition.
auf dem Bürgersteig is an adverbial prepositional phrase of place—an optional modifier giving location, not an object of the verb.

Why is die used before Passagiere? In English you might say “Passengers waited …” without “the.”

German normally requires an article or another determiner before common nouns in statements. Bare plurals without an article are very rare in neutral prose (you might see them in newspaper headlines or lists), so you use die Passagiere for “the passengers.”