Learn French: Vous pourriez me montrer un passeport valide ?

Pourriez is the conditional form of pouvoir (can / to be able to), so it translates to "could". Just like in English, asking "Could you..." is much more polite and less demanding than asking "Can you..." (Vous pouvez...). This is the standard way to make polite requests in French.

Why is me placed right before montrer?

In French, object pronouns like me (to me) go directly before the verb they belong to. Because you are asking someone to show to me, the me belongs to the infinitive verb montrer (to show), so it sits right in front of it.

This sentence has the word order of a normal statement. How do we know it's a question?

In spoken French, it's very common to ask a question simply by using normal sentence order (Vous pourriez...) and raising the pitch of your voice at the end. You could also use inversion (Pourriez-vous me montrer...) which is a bit more formal, but the version here is perfectly natural and polite for everyday speech.

Why does the adjective valide come after passeport?