Learn Russian: Мимо чьей машины мы проехали?

Because of the preposition мимо (past/by). In Russian, мимо always requires the following noun and its modifiers to be in the genitive case. Here, both the feminine pronoun чья (whose) and the noun машина (car) change to their genitive forms.

Why does the preposition мимо come at the very beginning of the sentence?

In English, we often leave prepositions at the end of a question like 'Whose car did we drive past?'. In Russian, you cannot strand a preposition at the end of a sentence. It must stay directly in front of the noun phrase it connects to, so the phrase meaning 'past whose car' (Мимо чьей машины) stays together at the start of the question.

How do you form the word чьей from the dictionary form of 'whose'?

The dictionary form is чей (masculine). Since машина is feminine, the base form here is чья. To put чья into the genitive case, the ending changes to -ьей, resulting in чьей. This is a special pronoun declension pattern that is good to review at this level.

Why use the prefixed verb проехали instead of just ехали?