Learn Spanish (Spain): Querría renovar mi pasaporte caducado. - I would like to renew my expired passport.

Querría is the conditional tense of querer (to want). Using the conditional softens your statement from a direct demand ("I want") to a polite request ("I would like"). This is the standard, respectful way to ask for a service in Spain, such as at a police station or embassy.

Could I say me gustaría instead of querría?

Yes, me gustaría is also a polite way to say "I would like." However, querría is often used when you are stating a direct intention to get a transaction or service done. Me gustaría can sometimes sound slightly more like a wish or a hypothetical desire. Both are perfectly polite, though!

I know querer has one 'r' in the middle. Why does querría have a double 'r'?

Querer is irregular in the conditional (and future) tense. Usually, you form the conditional by adding endings to the full infinitive. But querer drops the 'e' from its ending, mashing the two 'r's together to form the irregular stem querr-. That gives us querría instead of the incorrect querería.

Is caducado a verb or an adjective in this sentence?

It is acting as an adjective here, but it is formed from the past participle of the verb caducar (to expire). Because it is describing pasaporte (a masculine, singular noun), it must agree in gender and number, remaining caducado.